You don’t have to shower after a hot spring, however most people do—at least when shower facilities are available. In some remote springs, there aren’t any showers to use!
If you’re visiting the Blue Lagoon, the Sky Lagoon, or another geothermal spa, you’re requested to follow the etiquette of showering before you visit. By the way, it can be a particularly good idea to wash your hair after using the Blue Lagoon, as the silica in the water can make your hair quite dry.