Slide a plastic putty knife around the edge of the stain, and scrape it off. The plastic knife should not gouge the closet door, and be careful not to scratch or damage it.
Lay a hot, damp cloth over the stain to loosen any remaining nail polish. Hold it in place for at least 10 seconds, and then scrape it with the plastic putty knife some more. Continue heating the cloth and warming the nail polish until most of it has been removed.
Rub 0000-grade fine-steel wool on the stain to remove any lingering nail polish. Only use this process if you have enough remaining nail polish to be visually unappealing. The steel wool may remove a bit of the finish, depending on what the closet door is made from.