If the exhaust pipe is damaged can it cause the car to die and not want to start?

If it is restricted enough, the exhaust valves could burn out. Other than restriction, I wouldn't see how. Only if the pipes are pinched shut or almost completely blocked. Theres no backpressure it will stall out constantly or not start at all. (ever paint a potato black and stick it in someones exhaust pipe?) Theoretically, you can run the car with a huge gaping hole or no exhaust at all, it will sound like junk but it will run.