No, it's not critical. You can make an oven entirely out of mud / cob if you wanted without using any firebricks. The primary reason bricks are ideal for the base is because they are clean and will give a nice crisp cook to the bottom of the pizza. Firebricks are preferred over regular bricks because they can handle higher heats and wont crack. Similarly with the inside of the dome it's preferable to use fire bricks as they'll last longer with the high temperatures. For the outside of the oven you can use regular bricks or cob because it wont be exposed to the high heats inside the oven.
What about using ceramic tile for the floor?