1) Can you find out what kind of wood the drawers are made from? 2) And since it's possible that the drawer can be made out of MULTIPLE types of wood, can you 'smell' which wood is offing the odor? (What kind is that?) 3) Do you know if the smell is the wood, itself, or if the smell is from something that was 'applied' to it?
I'd hesitate on painting or polying the inside of the drawers, or the outside part of the wood that goes inside the drawer as well UNLESS the drawers are on rollers. If the drawers simply operate on wooden tracking, then almost ANYTHING you apply to any of the wood will 'gum up' the works, and give you nothing but problems when it comes to opening and closing them.