Will A Starter Cause A Battery To Go Dead?

I assume you mean will a bad starter cause a battery to go dead.  Depends on the circumstances.  Certainly trying to start the car for a prolonged amount of time will draw the power from the battery to the point that it will no longer have enough amperage to turn the engine over.  But that is not a dead battery.  I consider a dead battery to be burnt out and won't take a charge.

If you mean, will a defective starter drain the power from the battery, say while the car sits overnight.
That would be very unlikely.  The starter can only draw battery power when the solenoid (heavy duty switch) is closed.  The solenoid only closes when the low amperage current from the ignition switch is applied.  So, you would need at least two of the three components (ignition, solenoid and/or starter) to be defective or I guess a bad solenoid might act on its own, but it would very possible cause a fire.

If the car is two or more years old or if you purchased the battery 4 or so years ago, it's probably a bad battery.