Cats and dogs do. Even small mammals sneeze. Do only mammals sneeze? Do they all? Whales?
Iguanas do, as a way to expel excess salt from their bodies.
National Geographic is on the case. According to the cite, insects probably do not sneeze.
But do all mammals?
Whales don’t sneeze, and they’re mammals, so… no.
It’s a little more nuanced - Can Whales Sneeze? | Whale Facts
My guinea pigs would occasionally sneeze. It was a tiny little “chuh” sound. 
Rats sneeze. In fact they are fairly prone to respiratory infections, so excessive sneezing is one of the signs that it’s time to start a course of antibiotics.