Clean out your dresser by removing any clothing, accessories and all other items you currently have inside of the drawers.
Remove each of the drawers from the dresser completely to allow each part of the furniture to air out.
Sand the insides of the dresser drawers lightly using sandpaper or a portable hand sander and vacuum up the remaining shavings of wood and dust within the dresser.
Fill the dresser drawers with various pieces of newspaper to help absorb all of the moth ball odors within them.
Use a washcloth that is lightly dampened with vinegar to wipe the remaining areas inside of the dresser drawers and let each one air dry before placing the drawers back into the dresser to use again.