How do you remove musty odors from wood furniture?

Completed wash out the furniture with white vinegar.Allow to dry and put open boxes of baking soda.deactivated charchol or coffee grounds in it.

I washed down my smelly furniture with a damp cloth of vinegar and then put Cling Free in the drawers, cabinet. It has worked out well.

Place a dish of charcoal in each drawer for several days. I would also wipe out the drawers with a rag soaked in rubbing alcohol.

Put it outside in dry sunny location. Do not let it get wet (rain, dew, high humidity, etc). Return it indoors at night and put it out again on dry days and check progress on reducing musty odor. Remove drawers and expose them to sun and air. Keep them off ground. Rotate draws and carcass of chest to expose it to sunlight and air. Polish the finished wood with appropriate furniture polish or oil. Wipe down painted wood with appropriate cleaner with mold and mildew cleaner. Use anti-dessicant to absorb moisture and odor from interior of drawers. Use dry herbs and sachets in drawers to give off pleasant odors once the piece is returned indoors and draws are reinstalled.

Use a dehumidifier in your basement to reduce humidity during warm weather to help preserve items stored in basement. Excess humidity can encourage mold growth and breakdown of structural members of the building.

I would put it in the sunshine for a few days, taking it into a protected area at night..after a couple of days, it should smell better!