In the PHB, on page 193, the Use an Object action is described:
Use an Object
You normally interact with an object while doing something else, such as when you draw a sword as part of an attack. When an object requires your action for its use, you take the Use an Object action. This action is also useful when you want to interact with more than one object on your turn
But what exactly is considered an object?
More specifically:
Is an arrow an object? Are weapons, armors, and shields objects?
An object is any physical thing that is not a creature:
... targets creatures, objects, or a point of origin for an area of effect...
Chapter 10. Spellcasting, pg. 204 of PHB
The above description lists creatures, objects, and points in space as the 3 distinct, mutually exclusive types of targets. Note that spell effects are not physical things, and so are not objects, creatures, or points in space, though spell effects can affect, create, or share space with one or more of them.
A creature is anything that can take any (or at least most) of the general actions. All creatures fall into the categories of either Player Character or Monster:
A monster is defined as any creature that can be interacted with and potentially fought and killed... the term also applies to humans, elves, dwarves, and other civilized folk who might be friends or rivals to the player characters.
Introduction, pg. 4 of MM
So, you can also say that anything besides the PCs that isn't covered by the MM is an object.
This means that arrows, weapons, armor, and shields are, in fact, objects. NOTE: using some objects have more specific actions, so the "Use An Object" action you quoted does not apply. This is case for arrows (ammunition property of weapons and the attack action), weapons (the attack action), and armor and shields (donning and doffing and armor class). If you do something unconventional with one of these objects, such as using a sword as a lever to crank a wheel, the "Use An Object" action would apply.
From DMG p. 246 (which, by the way, is where Crawford tells us to look at if we want to see "What counts as an object in D&D") or here in the DM's basic rules:
For the purpose of these rules, an object is a discrete, inanimate item like a window, door, sword, book, table, chair, or stone, not a building or a vehicle that is composed of many other objects.
All the items listed are objects, as an arrow is a discrete inanimate item, as well as a weapon, an armor or a shield.
This definition gets fuzzy when taking into account magical items with sentience, as they are not inanimate any more (but they are still objects), but we do not have a canonical answer then.
I would also like to note that, while this is the game definition of the term, it doesn't seem a good one, and sometimes relying on common sense would be better. As mentioned in the responses to the linked Crawford's tweet, a book consists of many other objects (pages, cover), as well as clockwork toys...
The answer:
The Use an Object action is a catch-all that should be used generically when no other rule or action can be taken to interact with or otherwise use an object.
The reasoning:
The definition of object is defined on pg 246 of the DMG:
...a discrete, inanimate item like a window, door, sword, book, table, chair, or stone, not a building or a vehicle that is composed of many other objects.
However, the rules given on pg 246 pertaining to objects have to do with damaging an object and make no mention of Use an Object -- not to mention the definition here is slightly befuddling as it attempts to create a distinction between a building and parts of a building; but doors can have windows, and doors can have locks and hinges, and traps, and other mechanisms. The point being, this is a somewhat flimsy definition and doesn't change the rules for the "Use an Object" action. For simplicity, let's stick with "a discrete, inanimate item" for now.
5e D&D's rules are exception based: a general rule always applies unless a more specific rule applies. This can be applied to all objects. Since "object" is a class that encompasses all (inanimate) things, "Use an Object" is the generic action you take when you want to interact with an object that doesn't already have some other rule for interacting with it.
For example, you wouldn't Use an Object to attack with a weapon, because weapons have a more specific action for their use, the Attack action. You could, however, Use an Object to interact with a weapon in a non-attacking manner, such as drawing it from a sheath. The rules also do specify that you get one free "object interaction" per turn,
You can also interact with one object or feature of the environment for free, during either your move or your action. For example, you could open a door during your move as you stride toward a foe, or you could draw your weapon as part of the same action you use to attack.
which, as stated, can be used to draw one weapon as part of a move or attack action, but if you were to draw a second weapon in the same turn, that would require the use of the Use an Object action.
Note that this only applies to the generic Use an Object action, and is limited to things you could reasonably do in "less" amount of time than a regular action. Drawing a weapon or opening a door while you move is reasonable. Tying a rope around something is probably not. This will likely vary from DM to DM.
Using a magic item, similarly, has its own rules associated with it and does not normally use the Use an Object action.
Arrows, weapons, armors, and shields are all objects, but if you read the rules for these specific types of objects you will find they all have specific rules for how they are used. To "use" armor (that is, to wear it), you must take time to put it on, and that time varies from armor type to armor type.
Shields are a type of armor, and all armors have rules for donning and doffing: it takes one action to don a shield, and 10 minutes to don heavy armor. Donning armor is not Use an Object because it has a specific rule on how to use it.
Arrows have their own rules as well:
Ammunition: ... Drawing the ammunition from a quiver, case, or other container is part of the attack (you need a free hand to load a one-handed weapon)...
So when you use an arrow, you're doing so for free as part of your attack (unless you're not using it to attack with, in which case, you guessed it, it's Use an Object).
All this to say that the Use an Object action is a catch-all that should be used generically when no other rule or action can be taken to interact with or otherwise use an object.
If you want to do something with them that (a) isn't part of another action, (b) isn't an attack, or (c) other rules say requires an action, the you need to take the Use an Object action.
During combat, on your turn, you can interact with one object "for free" as part of your movement or action:
You can also interact with one object or feature of the environment for free, during either your move or your action. For example, you could open a door during your move as you stride toward a foe, or you could draw your weapon as part of the same action you use to attack.
(PHB, p. 190, emphasis mine)
This covers many cases where arrows and weapons would be "used" in conjunction with an action.
If you want to interact with a second object, you need to use your action.
(PHB, cont'd)
So if you've already drawn your sword as part of your attack action, you can't draw a second weapon to make an off-hand attack on the same turn. Arrows and weapons would again be 'objects' for the purpose of this rule.
Some magic items and other special objects always require an action to use, as stated in their descriptions.
(PHB, cont'd)
Shields, per the Donning and Doffing Armor table on p.146 of the PHB, take an action (the Use an Object action) to put on or take off. Armor of any kind takes much longer, and each round a character is putting on armor, they would take the Use an Object action.
Use an Object doesn't include attacking with arrows or weapons, of course; that's what the Attack action is for.
The game gives this definition of an object (DMG p. 246):
For the purpose of these rules, an object is a discrete, inanimate item like a window, door, sword, book, table, chair, or stone, not a building or a vehicle that is composed of many other objects.
It provides further examples in the tables for assigning AC and hp to objects, (p. 246 and 247, DMG): a bottle, lock, chest, lute, barrel, chandelier, cart, window, and as materials: cloth, paper, rope, crystal, glass, ice, wood, bone, stone, iron, steel, mithral, adamantine.
The concrete asks are thus easy: weapon, armor, shields, arrows all are objects under this definition.
There are corner cases for which the definition is more murky:
Plants are a case for ruling, but probably are not objects. They could count as harmless creatures. It can be argued that normal plants must be objects, but this does not fit with "inanimate", because animate is defined as "alive, possessing life", and plants are alive.
Liquids are a case for ruling. Is a body of liquid an object? The majority consensus is no, but it is not waterproof. If liquids were objects, it should be possible to assign AC and hp to them, but there are no liquids on the material list for AC, and you can hit a body of water all day long without damaging it (in spite of Xerxes whipping the Hellespont, although you could declare this is due to immunity to such damage). The most appealing argument for treating liquid as an object is that a flaming sphere or fire bolt should be able to ignite a puddle of oil, which it cannot if the oil is not an object. But then many spells are unintuitive that way, a fireball does not evaporate water either.
Corpses are objects. It is generally established that a dead creature is an object. Some argue that it is both a creature and an object at the same time because it is referred to as a "dead creature", but that still allows them to be an object.
Connected things can be objects, e.g. a window or door normally is connected to a building and is explicitly an object. But they appear to be sufficiently distinct, to count as discrete objects.
Composed things are objects if not too large. The defintion says an object is "not a building or vehicle that is composed of many other objects" and this restriction exludes are very large things from being objects. Otherwise you just could true polymorph a castle into a frog. It does not mean objects in general cannot be composed. The cart example in the list of object sizes is both a vehicle (PHB, p. 156) and composed of wheels, axle etc, but is small enough. The lock is an example composed of an object that is composed of many parts, but small enough. In contrast, large structures, like houses or ships, are too big to be treated as a single thing for purpose of attack and damage, and need to be broken down into sections they are composed of. The DMG advises on p. 247 to split up Huge or Gargantuan objects: If you track hit points for the object, divide it into Large or smaller sections, and track each section's hit points separately. Where to draw the line here in the end is DM fiat.
Chests with content are objects, but a case for ruling. The arguments here are the same as for a lock: there may well be smaller parts contained within them, but if those are not separable and they presents as a single item to interactions, then they can be treated as an object. For example, you can lift a full chest with a single object interaction, as it will lift the enclosed contents. The same would work for Telekinesis. What happens if you teleport a full chest, magically shrink it or disintegrate it, is less clear-cut.
Magic items are objects -- there can be magic locks, chests, lutes etc., which just because they are magical in nature are still chests, locks or lutes. Even sentinent magic items are objects, they do not count as animate. Note that magic items are special, because to activate or use them, you cannot use the "Use an Object" action, they have their own rules for that.
Magical effects are not objects. Although effects like the sensor of a scrying spell, may well be discrete and inanimate, the sensor for example is described as a luminous orb about the size of your fist in that spell, fitting both criteria. However, magical effects are no objects. "Magical effect" is not on the list of examples, nor "force" or "illusion" on that of materials. Dispel magic says: Choose one creature, object, or magical effect within range, and if magical effects were objects, there would be no reason to list them in addition to objects.
Objects are inanimate, discrete, physical things. If they are very large, the DM needs to segment them into smaller sections that can be treated as objects for purpose of attack and destruction. The DM needs to make a judgement call in many cases, like for liquids or plants.
Dungeon Master's Guide, chapter 8, Objects section, pages 246-247 gives us the majority of what we need: a definition, examples for Armor Class and Hit Points, a suggestion for fairly large things, details related to damage types, and details on Damage Thresholds.
The definition for Objects reads (emphasis mine):
For the purpose of these rules, an object is a discrete, inanimate item like a window, door, sword, book, table, chair, or stone, not a building or a vehicle that is composed of many other objects.
Discrete generally means that something is easy to identify as separate from what is around it. So this would mean that you can readily tell where the thing in question ends, and everything around it begins. An example would be individual bricks in an uncovered wall, or a wall compared to its roof.
Inanimate simply means possessing no life, but due to the sometimes fuzzy nature of that in 5e, I think it would be logical to extend this to also possessing no choice. Basically, the inability to do anything without action of an outside force.
As for examples within the definition, I chose to highlight the book, because it is an item in the Player's Handbook, and we know that it, though made of multiple parts, has statistics only for itself as a whole, even though both parchment and paper are separate items themselves. Indicating that complexity doesn't matter.
Lastly, from the definition, we are told of buildings and vehicles, but specifically that they are not objects only if they are composed of other objects. If they were always not objects, nothing would be needed after the word vehicle. We even have an example of them as objects directly for the Objects section, as a cart is listed within the Object Hit Points table, and a cart is a drawn vehicle, as shown in the table Tack, Harness, and Drawn Vehicles on page 157 of the Player's Handbook.
Further, there is a table on page 119 of the Dungeon Master's Guide called Airborne and Waterborne Vehicles which features a number of vehicles with Damage Thresholds, something listed as present on some objects. Of note, the vehicles as listed here, or in the Player's Handbook, only have statistics for themselves as a whole, not for any parts.
Finally, there are 3 examples of items that can help us narrow down what it means to be composed of other objects.
The first is the magic item Daern's Instant Fortress (Dungeon Master's Guide, page 160). When deployed, this is a building where the roof, the door, and each wall, has its own hit points. These are all vital parts of the whole, without them, the building isn't a functional building, thus it is composed of these parts. So this building is not an object itself. The parts themselves are objects.
The second is the Keelboat as listed in Ghosts of Saltmarsh (page 188). This item not only features statistics for itself as a whole, but also for each vital part of its structure: the hull, helm, oars, sails. It also lists a ballista, but that isnt vital. If any of the vital parts is destroyed, the vehicle ceases to function, thus it is composed of them. So, the keelboat, as listed in Ghosts of Saltmarsh, is not an object. But each part is.
Last is the Demon Grinder from Descent into Avernus (page 219). Like Daern's Instant Fortress and the Keelboat as listed in Ghosts of Saltmarsh, this item has parts listed. Unlike those other items however, the listed parts are not vital to the existence of this item, so their loss would not operationally destroy this item. Thus, this item is an object.
Sorry for the long-winded response, here is an easy step-by-step checker to determine if something is an object or not:
If the answer is negative to the above questions, then the item in question is an object.
The DMG says:
For the purpose of these rules, an object is a discrete, inanimate item like a window, door, sword, book, table, chair, or stone, not a building or a vehicle that is composed of many other objects.
We are told an object is discrete and inanimate. "Inanimate" means that it is not a creature, which separates it from other fundamental game entities that interact with the rules in characteristic ways (objects, creatures, spell effects, dungeon hazards, etc.)
"Discrete" means that it can easily be recognized as separate from the environment and separate from other objects - which is where the complications begin.
Is a weapon an object?
We are told that an object is an item, "like" a window, door, sword, book, table, chair, or stone. Thus, the DMG says a sword is an object, but it also says objects are not composed of many other objects. A sword has at least three parts (blade, crosspiece, pommel) which were joined together by a smith. When the smith was handling these items they were each separate objects, but once joined together they became one object, the sword.
Is an armor an object?
The heat metal spell allows you to (emphases mine):
Choose a manufactured metal object, such as a metal weapon or a suit of heavy...metal armor, that you can see within range. You cause the object to glow red-hot. Any creature in physical contact with the object takes 2d8 fire damage when you cast the spell.
So yes, when targeted by a heat metal spell, a suit of armor is an object. And yet, real world plate armors consisted of at least twenty individual pieces which were each put on separately, which is why it takes so long to don and doff the armor. We also know that at least some of these pieces, such as the gauntlets and helmet, can be considered objects in their own right, since once you acquire magic versions of these (such as gauntlets of ogre power or a helm of brilliance) you can remove the normal pieces of your armor and switch in magic ones. So, if I am wearing a suit of armor, it is one object, targetable by a heat metal spell. But if I take off the helm and gauntlets and set them on the table next to me, the suit of armor then consists of four, separate and discrete, objects - any one of which - but only one of which - could be targeted by the spell.
PC's most typically interact with objects by using them as you are asking about ("Use an Object" Action, but also "Interacting with Objects around you" sidebar on PHB 190) or attempting to damage them ("Statistics for Objects" DMG 246). However, there are many more obscure possibilities, such as those of a 17th level Knowledge Domain Cleric performing an 'Object Reading' to "see visions of the object’s previous owner" (PHB 59). Whether or not something is an object itself, or is merely a part of another object, or is something itself composed of other objects, depends on the context in which the PC's are interacting with it. An object is some thing with which a game entity interacts. This is a functional definition rather than a categorical or linguistic one.
For example, if a party was attempting to get through a locked door quickly, the barbarian might attack the door itself, in which case the entire door would be considered an object he was damaging. But if they were instead trying to get through the door quietly, then the rogue might work at the lock while the artificer tried to disassemble the hinges. In this case the door would not be an object, but a collection of other objects, and the lock and hinges would be the objects with which the PC's were interacting.
As it turns out, the things that surround the PC's are in a sense Schrödinger entities: they are simultaneously objects and not-objects until the point in time when a creature or spell interacts with them, and then they become fixed as objects, at least for the duration of the interaction.
Ultimately, this goes all the way back to the First Three Rules about how the game is played. A player should not be asking "How many Objects do I see in the room?" or "What here can I do my Use an Object Action on?"
Rather, the DM describes the environment in narrative, naturalistic terms - not as a list of game mechanics entities.
The player responds with what they would like their character to do.
The DM narrates the results of those actions, perhaps noting whether it was a Use an Object Action, a free Object Interaction, or something else, as well as what Object was affected. The same thing the DM is calling "an object" now might not be one later, depending on how the characters later interact with it. For example:
PC1: "I hold my spellbook while I ritually cast my spell."
[PC1 is using the Cast A Spell action - and it doesn't matter whether the book is an object or not.]
PC2: "I knock the book PC1 is holding from their hands with my staff, rip out a page, and flee."
[The DM has PC2 make a Disarm attack, and treats the entire book as a single object, the target of that attack. PC2 is successful. When they attempt to tear out a page, though, the book becomes a collection of objects, with multiple sheets of parchment, many fibre cords or leather thongs, and two separate wooden boards. PC2 now uses their free Object Interaction to grab one of those objects, a single page, without applying the rules for damaging an object.]
PC3: "I shoot a firebolt at the page PC2 is holding."
[Here the page is a single object, targeted by the spell, and we are applying the damage to objects rule.]
An object exists at the scale of a game interaction, at the time that interaction occurs.
