A fire requires three things: a fuel, an oxidizer and enough energy to start the reaction. Oxygen is an oxidizer. However, just as not all fuels are variants of coal, not all oxidizers contain oxygen. In fact, oxygen isn’t even the best oxidizer. Fluorine is, and other halogens generally make good oxidizers as well.
So yes, if oxygen is replaced by another oxidizer, combustion can take place.
Most things that will burn in oxygen will also burn in chlorine and fluorine. Under the right conditions bromine and iodine also.