Sounds like the nail polish remover partially dissolved the finish on your nightstand and when it dried, the napkin stuck to it. you can release the bits of napkin with the same nail polish remover. Use a q-tip moistened with the nail polish remover and carefully apply to redissolve the finish. The bits of napkin should come right off.
Unfortunately, you have marred the original finish on your nightstand. If it is a polyurethane finish, you should be able to touch it up. The same goes for a varnish or shellac finish. If you don't like the results, you can always strip the finish from only the tabletop and apply a new finish. Good luck with your refinishing project.