If you run out of fuel and try to keep going or crank the motor hoping it will start you could burn up a fuel pump. The pump uses the fuel as a lube and/or coolant so if it is dry and operating... buh-bye. If the fuel pickup is clogged, same thing. The fuel filter should be an easy check though. I would pull a hose off at the engine block and turn the key to see if fuel is at least getting to the motor. If it is, then the problem is more than likely up front.
What year is this Firebird? An older 'bird could have a mechanical pump on the block or an electric on the frame rail. You would generally hear an electric if it were on the frame rail though, in my experiences. Some cars do have an access to a tank mounted fuel pump from above so you don't have to drop the tank to inspect things. It sounds like you may want to thumb through a manual or find another forum to ask more informed 'bird people though. It could be a number of things. Good luck.
Sent from my "two hands on a keyboard"