I usually unplug the bulb, light never comes on again.
alternator/battery light comes on, the alternator's gonna stop working if it already hasn't. at that point, you're running on battery. at 10 volts, things start getting "choppy". at 9 volts, the ECU shuts down, which means the car shuts down.
on OBD2 cars, pulling the check engine light will prevent it from passing most state inspections.