How can you remove mold and its smell from wooden furniture?

OZONE will do the trick. Get your hands on an air purification system that generates ozone and your problem is fixed. Ozone will get rid of any odor known that I am aware of.  

Answer

The other poster is correct, but with one exception ... mold thrives on moisture and you are trying to save some furniture. This is what you do:
You have to clean your furniture with 25% bleach in water. This will kill the mildew and mold and enable you to remove it with some soap and water. WEAR RUBBER GLOVES AND BE IN A WELL VENTILATED AREA WHILE DOING THIS PROCEDURE. Some discoloration may remain in the wood. THE SOLUTION IS CAUSTIC, BE CAREFUL WITH IT!
2/3 cup of trisodium phosphate, 1/3 cup of detergent, 1 quart of household bleach, and enough water to make a gallon.
NOTE: If the job is small please cut this back by more than half. DO THIS OUTSIDE AND AWAY FROM PETS AND CHILDREN. AFTER YOU HAVE FINISHED HOSE OFF ANY AREA YOU HAVE BEEN WORKING ON.
Then when the wood is perfectly dry give it a good coat of polyurethane varnish. You can go into any large Hardware store that sells building supplies and ask for their advice on stains for the wood. You'll have a great new piece of furniture before you are done.
Here is a bonus for cleaning out odors in drawers: Springer coffee grounds (dry) inside the drawers or use kitty litter for 2 days. This will not only soak up any traces of mildew, but also get rid of the odor. Musty smells also vanish.