Would the pressure release valve going bad on a water heater cause the water heater to quit working?
No, a water heater will continue to heat the water with a bad TPR valve. A failed TPR valve is a safety issue since the water heater could heat the water to the point that the tank explodes and does significant damage to your home, roof, and probably the home of your neighbors. Several other components would also have to fail for this extreme situation to happen (the thermostat would have to heat beyond the normal cutoff, a check valve preventing water going back into the city water lines needs to exist, and no other valves in the house can be leaking) so it's fairly rare. But with the danger to your life, it's still a good precaution to ensure this valve is properly functioning.
if you tpr fails the valve closes and the tank gets no oxygen to fire. it will run long enough till the oxygen is depleted. time to get a new tank. thermal protection fuses are very hard to find, you can swap out the tank quicker