Can magical spells actually damage objects?
So something standalone, e.g. the pile of wood over there in the corner of the room?
And on a character, e.g. the quarterstaff in a character hand?
[I think they can, but I'm concerned I'm confusing DnD 5e with earlier versions]
The DMG (pages 246-247) has specific information about how objects can be damaged including ACs for different substances and suggested hit points based on size and resilience. These rules can easily be applied to damaging spells too.
I'm not going to reproduce the whole entry, but your pile of wood would probably have an AC of about 15 and hit points of about 3d8 (if it's loose branches) or 4d8 (if its cut lumber).
Please bear in mind that many spells specifically state that the target is a creature. I would probably disallow such spells from being used against objects (unless they've been animated and are attacking the party!).
There are a couple of entries which are quite important though.
Given enough time and the right tools, characters can destroy any destructible object. Use common sense when determining a character's success at damaging an object.
and...
Objects are immune to poison and psychic damage. You might decide that some damage types are more effective against a particular object or substance than others. For example, bludgeoning damage works well for smashing things but not for cutting through rope or leather. Paper or cloth objects might be vulnerable to fire and lighting damage. A pick can chip away stone but can't effectively cut down a tree.
Some spells specifically state that they damage objects.
Fireball (PHB 241) "ignites flammable objects in the [chosen] area that aren't being worn or carried." To my knowledge, none of the books cover objects burning, but it's reasonable to assume that a Fireball would ignite a wooden hut, cause lingering fire damage in an area, and eventually turn it to ash.
Shatter (PHB 275) states that "A non magical object that isn't being worn or carried also takes the damage if it's in the spell's area." For this, I would use the rules for damaging objects (DMG 246-247) to determine hitpoints.
Note that both of these spells prevent worn or carried objects from being damaged, so a wizard's quarterstaff is generally safe.
Some spells can damage objects.You can use Fire Bolt to attack the pile of wood over there in the corner of the room. However, most of spells only can affect creatures. For example，Fireball can ignites flammable objects but can not damage them directly, but Fire Storm and Fire Bolt can both ignite objects and damage them.
