In Italy cheese is absolutely not required, even if Margherita is the most common base so most pizzas have cheese. Even so, there are many types of pizza with no cheese, and many others with no tomato.

It's also very common to see pizza farcita, which you can imagine as a pizza sandwich. For example a very common one is "pizza e mortazza" in Rome, which is a pure puzza with no topping but filled with mortadella (a type of ham). But various kinds of fillings are possible

Another example would be focaccia, most of them don't have any cheese at all.

There is even sweet pizza with no cheese, for example pizza with Nutella