Peel it up, as much as you can. Then use some commercial Goo-remover product, to soak the last glued down bits with. How come you are worried about the finish, if you are gonna cover it up again?
Cause if you dont care, then the easiest way is to use a single edge razor blade scraper, scrape it all out, use a solvent to get the last of it, then just respray with a spray bomb in brown wrinkle (or red, or whatever).
Kennedy sells touchup paint, but if for some reason you dont want to buy from them, you can get brown wrinkle spray paint.
Then go to the nearest fabric store, and find a felt you like, in a pleasing color or pattern- NFL logos, say, or Halloween. No, really felt is usually only available in solid colors, but any fabric will work just as well. Cut fabric to size, test fit, then use some spray glue on the drawer, and put it in. Trim any over size with a mat knife or another single edge razor blade.
The goal is just a soft bottom to keep tools from banging each other up- You could use red velvet if you wanted.