Why is the starter staying on when the ignition is off?

Answer

When a starter continues to run after the starter switch is disengaged, it could be caused mechanically by the pinion gear sticking in the flywheel teeth ( with multiple causes- bent starter shaft, worn flywheel teeth), or electrically: stuck starter relay (if it uses one), stuck starter solenoid (either by the metal piston sticking, or the electrical contacts sticking), or of course the ignition/starter switch itself. One needs to eliminate the possible causes one by one to diagnose the cause.