I think it might depend on where you come from, but they would most commonly be referred to as 'drawers' where I live.

It's common to call it a 'dresser' even though it is not. 'Bureau' is an older less commonly heard term for a chest of drawers.

I'm a native U.S. speaker. My wife and I have pieces of furniture with drawers for storing clothes, and we call them "bureaus." Elsewhere in the house, we have "cabinets" which usually have a variety of storage compartments, including drawers. We also use the phrase "set of drawers" (not "chest of drawers.")

Perfectly acceptable to just call it a 'chest' as well.

Coming late to the party here, but for me, the answer is 'No'. The fixed compound is 'chest of drawers' - this is the name of the piece of furniture, and it cannot be shortened. 'Drawers' are not a item of furniture, and a 'chest' is a different thing entirely - so you have to say the whole thing or not at all. By all means, use 'dresser' if you're talking to American English speakers, but bear in mind that this word may confuse British English speakers. In fact, I didn't know what a dresser was, so I had to google images of 'dresser' to make sure it was the same thing as what British English speakers call a chest of drawers. 'Bureau' is also likely to confuse - if you google 'bureau', you 'll see it's a fold-open desk with drawers beneath.