Not sure which direction to go. Inclined to think plural, because it's usually thought of in full as a "chest of drawers", right? Though it could be a single drawer pulled from the larger piece of furniture.
I'd be tempted to keep it singular "drawer" simply because it maintains unambiguity, whereas "drawers" has a completely different meaning (see post #625503).
That word sense may not be a tag we use, but there's not much point in causing potential semantic confusion if we don't have to.
As for the piece of furniture, I'd be more likely to call it a dresser which is a tag we have in use, though there isn't much in it ATM.
I'd support using just "drawer" for just the drawer, but for the furniture I'd support "chest of drawers." If you really want to call it a "dresser," I think we need to call it a "bedroom dresser" as there are other pieces of furniture that can be called a dresser as well.
I think chest_of_drawers is fine too, though I never use that term, and at least in my dialect of American English, I'm not aware of anything else furniture-wise "dresser" would refer to. If we use "chest of drawers", I think an alias from "dresser" would be nice.
Wikipedia's dresser (furniture) is something I've only rarely ever seen and probably never needed to describe. If I needed to, I'd probably call it a sideboard. If it were glassed in, I'd call it a china cabinet.
Alright, I've aliased drawers to drawer (of someone does in some rare instance use it for clothing they'll get the point or someone else will fix it). And I guess we can go ahead and use chest_of_drawers, which I'll alias dresser to. Since I doubt we'll ever use dresser for anything else.