How do you get the new wood smell out of dresser drawers?

Try putting dryer sheets in the drawers. Also, fragranced soap bars, or drawer liners work effectively as well.

To eliminate the odor rather than just cover it up, use something like AFM Safecoat. It is a water-based sealant. It will not stain the wood, just cover it with an invisible coating. I would use this on the inside of the drawers and see if that helps. You may need to use two (thin) coats. This will keep the gasses that are causing the odor from escaping from the wood.

By new wood smell, do you mean the chemical off-gassing from the glues in plywood/particle-board furniture? If so, the lighter-weight, but more expensive, real wood dresser might be a better bet. Off-gassing can take a year or more to diminish beyond noticeable smells. This is why the people displaced by Hurricane Katrina who are living in brand new mobile homes now have lung problems.

I have had good luck removing odors with an ozone generator. They are awesome!!