Place one slice of white bread in a bowl, and cover the bread with white distilled vinegar. Place the bowl inside the wood item if possible. For example, place the bowl in a drawer if the item is a dresser or cabinet, and close the drawer. The bread-vinegar mixture will absorb much of the musty odor. Remove the bowl after 24 hours.
Wash varnished areas of the wood with wood soap by following the wood soap manufacturer’s package instructions. Dry the washed wood thoroughly.
Fill a bucket with 1 gallon of water and add 1 teaspoon of household liquid bleach, or use equal parts of water and lemon juice. Paint all unfinished surfaces of the wood with the solution. That includes inside and under drawers, the item's back and bottom and internal wood guide rails. Allow the piece to dry thoroughly.
Paint a thin layer of varnish over the unfinished wood areas you brushed with bleach or lemon solution. Dry the wood thoroughly. The varnish will seal in remaining musty odors.