First, wash the piece thoroughly and dry.You might use vinegar & water as it is a natural deodorizer & sanitizer. On an inside drawer you might resort to a dilute bleach water, but I wouldn't use on the finish. You can try the charcoal (this one works well for me), kitty litter, newspaper, soap (smells so good) etc.
What worked for me with an old trunk was to put it into the sunlight for a few days (bring it back in at night). This can make the wood darker (if a light colored piece), or lighter if a dark piece. So..if the smell is not from the top of the piece, I would cover it. Good luck with it.
Can you pinpoint the exact area the smell is coming from? There's probably no 'delicate' way to do this - you'll just have to get down on your hands and knees and sniff it! Its highly unlikely to be human urine so my thoughts were maybe an animal had 'marked' on it, e.g. a dog or cat. If you can find a spot near the floor, or even a drawer, where the small is strong you could try a product from the vet for getting rid of pet urine smells. If the smell is generalised all over the furniture it is either the wood, (not likely if it is sealed/polished - although maybe the internal carcass of the piece isn't sealed), or the finish. An expert furniture restorer may have come across this problem before and refinishing the piece may be the only answer.
If it is a valuable piece of furniture, you could call an antiques dealer for suggestions. If not, use lemon oil liberally and let it sit for a half-hour to an hour, if it is made of wood. Then wipe off. If the odors are in the drawers, try the newspaper method or take out the drawers and wipe lightly with a scented oil or as suggested -- air out on a sunny day for a few days. Good luck!
Stuffing the drawers with newspaper and waiting a few days might help. The newspaper should absorb the odor. If it's warm where you live, you might try setting it in the sunshine to "air out".
There are lots of things to try but one that might work is lots of charcoal briquettes in the drawers etc.
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