Get do I get the musty odor out of a piece of furniture?

You can sprinkle it with baking soda and let it set. Then vacuum it out. It may take several applications to get it all the way out.

I think you hac sprinkle Bi-Car of Soda on it. Leave for a few hours and vacuum it off. Worth a try.

ODOBAN works great. Take it outside spray, leave in the sun. If you are re-doing the piece of furniture, check out my post:

https://www.hometalk.com/diy/clean/furniture/how-to-get-rid-of-furniture-odors-before-refinishing-39720142

and for more details:

https://karupp-did.net/sorry-for-the-stink/

I have cleaned furniture with white vinegar.