Start by opening all of the dresser drawers and leave open for 24 hours and then fill small bowls with 1/4 cup of baking soda and place them inside the drawers. Pour some vinegar and water using a spray bottle to eliminate any remaining musty odor. For more information, check out http://www.ehow.com/how_6469769_musty-smell-out-chest-drawers.html.
have you made sure there is no mold? check all joints and seams of the dresser. if it is painted it can still be treated. mold needs bleach treatment. blot affected areas with 10% bleach solution and allow to dry. air for 24 hours to eliminate bleach odor after drying. if no mold then as sheilawangui says will work. ammonia sprayed on will work too if you need something stronger. *hint, for a room freshener, a bowl of ammonia placed up on the 'fridge keeps the kitchen fresh smelling.