If you constantly run your car with a dead battery will this hurt your alternator?

Yes, alternators are designed to maintain the battery, or recharge a battery that has been slightly discharged from starting the car. When the alternator is used to recharge a dead battery, it causes the alternator work harder and longer than it is designed for, causing it to overheat, thus causing damage to the alternator. A battery charger should be used instead. They are designed for charging dead batteries. After the battery is charged, let the alternator maintain the charge.