What can I do about a smell in my oak furniture piece?

It is probably mold- I suggest using " Simple green ",clean with a disposable rag, then rinse well and when dry ( a full day) use a good wood polish with citrus oil to help

protect the wood and smell better. You can add a sachet of dried Rosemary, Thyme, or other dried spices you like( you can have a combination of floral and citrus too).

I suggest a good rub down with TSP. Cheap, effective.

Takes away a lot of mildrew spots and dirt and just plain crud.

Sunshine also helps.

Ive heard coffee grounds (in a container) helps.

Drawers, Ive used sprinkling of baking soda, then vacuum out.

After cleaning it good you could use baking soda, charcoal or coffee to draw out the smell most times

You can also sand the wood lightly to remove top "layer" and put charcoal briquettes in container or bag and shut drawer for a few days. It draws out odors and dampness.

Also try to give it as much "air" as possible, it might take some time for the smell to go away, but it will!

Use orange oil. It will clean it up , condition the wood and kill the mold. Good stuff .

You could also use a vinegar mixture and leave it outside for a while. Maybe even varnish it. Use coffee grounds and rub them into the wood. I will sometimes even put a softener sheet in the drawers to help with musty odors. Just change them out when necessary.

I have treated it with vinegar water and lemon oil . I will do it a few times and see its certainly shining