Your dresser will need refinishing. If the finish is a tinted lacquer you will need to get a pro to do it, lacquer isn't easy to work with and must be sprayed.

If you want to try this yourself you will need to sand the affected down to raw wood to get rid of the oil before applying the lacquer finish. Another option is to sand, prime and paint with a good oil based enamel. It won't match exactly but would be a cheaper fix.