Charcoal, coffee, baking soda and cat litter are all effective in removing odors from musty furniture. Apartment Therapy recommends placing small briquettes of aquarium-grade charcoal on and inside of furniture for up to two days for it to absorb off odors. Discard charcoal after use, or store it for another application in the future.
Coffee can be just as effective as charcoal in sucking up odors, but because it can get stuck in upholstery, the treatment method is different. Apartment Therapy recommends placing ground, dry coffee in bowls on and around the offending furniture for 24 hours to allow the grounds to soak up musty odors.
An open box of baking soda is another effective odor killer. Place a box near the furniture, or for particularly musty furniture, sprinkle the powder inside cushions or in other spots that are not visible to fight odors fast. Use cat litter in the same way as coffee, setting it in small bowls on and around the furniture to soak up odors. Look for natural clay cat litter for the best odor-fighting results.
Apartment Therapy also suggests using oil-based cleaners, such as Murphy's Oil Soap, to remove odors from both wood and upholstered surfaces. Follow directions on the bottle to mix a solution to help strip odors from musty furniture.