I would spray and scrub with bleach. then when thoroughly dry, spray with vinegar to get any remaining odor out.
There is a liquid spray called Odoban...got cat smell out of plywood subfloor...might work.
If you are painting the furniture a color, but leaving the inside natural, then following very thorough cleaning and drying of all the interior and rear areas, coat every surface with two or three coats of flat finish polyurethane. The paint you use for the rest of the piece should also work to cover (after thorough cleaning and sanding) and remove the odor. Worked for a friend of ours who couldn't sell her house because of cat odor (two males who sprayed to mark territory). It was a nice house, with a sickening odor. She even removed her wall to wall carpeting to send it out to get professionally cleaned and had her floors cleaned thoroughly. She was so used to living with cat odor, she couldn't really detect if she was making progress (she wasn't!). Then I had heard on a real estate radio show that people who had cat odor had a big problem, but poly was the only way to get rid of it. So, following thorough cleaning to her wooden floors, she again had the carpets removed, had the areas coated with poly--the 2nd time she put down the freshly cleaned carpet, the smell finally was gone. People also use lumps of charcoal to absorb odors, but I don't know if that works for wood.
I swear by ODOBAN (available at Sam's Club). I use it on litterboxes, pet beds, laundry, pet accidents, even mop the floors with it. In addition to removing odors, it's a disinfectant and kills 99.99% of germs. I help with a cat rescue and couldn't live without it! BTW, I have no monetary interest in the company-just a very satisfied customer.
Make sure that you do the entire dresser, every last inch . Myself, I would not put clothing in it, only items that cannot absorb smells. I love to reuse but I would be afraid if anything got missed as you can get a diseases from mice feces and urine such as Hauntavirus. I would toss it. Its not worth the risk.
I agree with Odoban also, I have used it for years and love the nice clean smell after using it. also get mine at Sam's Club. Judy and Donna are pot on!
After I scrubbed mine, I sealed the inside of it. Then painted it. I used kilts in the inside, and the drawes.
If you are taking it down to the bare wood, scrub with a solution of bleach and a tiny bit of dish soap. Rinse well and let it sit out in the sun to dry. (Don't soak the wood, just a scrub.)
Then a coat of KILZ. That should do it.