I know a few people who travel around Belize with their dogs, so you can do it. A lot of hotels would probably accommodate you, assuming you maintained close control over the dogs. However, I wouldn't say it's common, as most visitors don't bring their pets, and most Belizeans (there are exceptions) think of dogs more as guard animals or watchdogs than as indoor pets.
A lot of hotels and lodges keep dogs to help in security, and there are quite a few homeless or stray dogs around, not to mention things like snakes, scorpions and killer bees, so you'd have to keep a close watch on your animals.
--Lan Sluder