According to Smithsonian reporting on this study, Ollie Jay at the University of Ottawa’s School of Human Kinetics found that by drinking a hot beverage, you effectively increase your body temperature, which causes you to sweat. However, as the sweat evaporates, it cools you down. Any tiny increase in body temperature from the hot drink is far outweighed by the greater decrease in body temperature from the evaporating sweat, leaving you feeling cooler than before.
The caveat, though, is that the cooling effect is only noticeable if the extra sweat can evaporate. If you’re wearing little clothing on a hot, dry day, give it a try. Since I’m in a humid locale, I think I’ll stick with the ice water—for now.