How do thrown weapons interact with the Extra Attack feature? Can you throw more than one weapon per turn?
How about two-weapon fighting? Do you need the Dual-Wielder feat to throw two weapons every turn?
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Throwing multiple weapons in a turn is limited by the action economy. The rules say you can only interact with 1 object for free on your turn (PHB 190). Any other object interactions require a full action.
This is different from grabbing arrows/bolts as bows and crossbows have the Ammunition property, which has the specific rule that grabbing ammo for these weapons is fee. Thrown weapons do not have the Ammunition property that grants this.
The rules for two weapon fighting specifically allow thrown weapons to be used for both the first attack and the bonus action attack.
So how many weapons can you throw? It depends on what you start your turn holding and the number of attacks you can make.
Extra Attack specifies that you can attack multiple times when you take the Attack action. Thrown weapon attacks are attacks, so they can be done more than once as long as you have the weapons in hand (or can draw them fast enough) and as long as you respect any special properties of the weapon that might limit that (for example, some non-throwing weapons have the Loading property).
Two-weapon fighting specifically allows thrown weapons for either of the two weapons, allowing you to throw two weapons in a turn.
Note that you can normally only draw one weapon in a turn (as your "interact with an object" option), so continually throwing two weapons per turn is unlikely. You can, however, consistently attack with a melee weapon and throw a thrown weapon every turn.
So, this is legalistic to the point of absurdity, but...
The Use an Object description (PHB p 193) states (emphasis mine):
You normally interact with an object when you are doing something else, such as when you draw a sword as part of an attack.
The description of the Ammunition property (p 146), which does allow for interaction with more than one object in a turn, uses almost the same wording (emphasis mine):
Drawing the ammunition [...] is part of the attack.
Note that the PHB distinguishes between "an attack" and "taking the Attack action" elsewhere (e.g. Extra Attack description, Two-Weapon Fighting). This sets up a direct contradiction with the box o' single object interactions (p 190). By Specific Beats General, the more narrowly defined case would hold, making weapons drawn as part of an attack (not the Attack action) the sole exception to the "one free object interaction per turn" rule.
While it seems silly to read so much into this, it seems even sillier that thrown weapons would be singled out as the only form of physical attack that could not be incorporated into a fully-functional build. Given how far the system appears willing to bend to allow quirky builds to be viable, it's hard to believe the rules would intentionally hamstring darts champions and axe tossers.
Personally, with items such as darts, I revert back to original DnD rules. There is no way it costs as much movement and time to handle 3 darts held in hand as it does to pull, nock, draw, aim and shoot an arrow. I have done both IRL and there's a massive difference. So the ability to throw 3 darts/round with ease especially as a proficient weapon… you have developed that automatic eye hand coordination to release darts with fair accuracy. And don't forget you still have to roll the hit.
(Note that this is heavily based my answer to this duplicate question.)
The only things that can be drawn as part of the attack are the ammunition used for weapons (such as crossbows) that have the ammunition property. From PHB (pp. 146-147):
Ammunition. You can use a weapon that has the ammunition property to make a ranged attack only if you have ammunition to fire from the weapon. Each time you attack with the weapon, you expend one piece of ammunition. Drawing the ammunition from a quiver, case, or other container is part of the attack.
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.
As you note, this means you cannot draw and throw three weapon in the same turn. At best, if you started your turn with two throwable weapon already drawn, you could then throw both of them (using Two Weapon Fighting), use your one free object interaction to draw another, then throw it (as your second attack as per Extra Attack).
Regarding house-ruling otherwise, see my related question here: Are there any balance issues with allowing thrown Javelins to be drawn for free like ammunition weapons?