Can you die if you wash your mouth out if you tasted nail polish remover?

It depends on what the nail polish remover is.

Most nail polish remover I'm familiar with is primarily a mixture of acetone, ethyl acetate, and/or 2-propanol. Ethyl acetate is pretty harmless; acetone and 2-propanol are a bit more of a problem, but "a tiny bit" is unlikely to kill you.

Some nail polish remover contains acetonitrile, which isn't terribly toxic itself but tends to be metabolized to cyanide, which is, well, cyanide. If you find yourself feeling ill afterward (breathing difficulties, slow heart rate, nausea, vomiting) you should definitely call the poison control center and/or head for the nearest emergency room. Cyanide poisoning is treatable, but not by anything you've probably got lying around the house.