Help Repair Chemical Damage to Dresser Surface
Just sand it smooth and repaint. It's really not that complicated. You'll have to be careful not to sand through the veneer, so do it by hand, don't use an electric sander. Start with coarse sandpaper, then medium, then fine. At the fine sandpaper stage, sand the whole top of the dresser, not just the damaged area, it'll help the paint adhere. 

Word of advice on choosing the paint color: you probably won't be able to get an exact match. So try to use a contrasting color instead, it'll look better than a color that almost matches. Or repaint the whole dresser at once.