I researched this topic once and found the same cheeses mentioned by Foccaciaman and Dankfoot. The most common choice I found was a combination of cheddar and mozzarella cheeses, but cheddar, mozzarella, "American", provolone and Jack are also used, quite often alone. And, sometimes, a combination of mozzarella and provolone, which is a common combination used by many pizzerias for a much broader range of pizzas besides cheeseburger pizzas. One caveat I found was to cover cheddar with another cheese on a pizza since cheddar apparently burns easily otherwise.
Surprisingly, I didn't find evidence of use on cheeseburger pizzas of Swiss cheese or the provel (St. Louis) blend of pizza cheeses (white cheddar, Swiss and provolone). Swiss and provel are both used on cheeseburgers, so I suspect somewhere someone has used both on cheeseburger pizzas. I just couldn't find where.
Some differentiators I found for cheeseburger pizzas include mushrooms, bacon, onions, pickles and jalapenos--on a tomato sauce/ketchup base or mustard or mayo (or both mustard and mayo). And, of course, one can add fresh tomatoes, onions and lettuce on the finished pizza. Basically these are all adaptations from a sandwich format to a pizza format.