Fill a spray bottle with undiluted white vinegar. Vinegar is an all-natural deodorizer that will remove the musty odor from the old furniture without the use of harsh chemicals.
Apply the white vinegar to the furniture liberally and allow it to dry on the furniture. Reapply the vinegar has needed. Vinegar has an unpleasant aroma but will dissipate as it dries and removes the musty smell.
Sprinkle baking soda liberally on the old furniture. Allow the baking soda to sit for a few hours--preferably overnight.
Secure the upholstery attachment to your vacuum. Remove the baking soda from the furniture with the upholstery attachment. The baking soda will remove any lingering odors trapped in the furniture's fibers.
Repeat the process if needed.