I have a 2001 Chevrolet Monte Carlo and I have the same issue! I would say it is either the coolant temperature sensor or the thermostat! Reason why I'm not 100% sure on which one caused the problem, I did them both since they were next to each other on the motor and it took care of the problem! :) Hope this helps you!
Also, the heater blower motor resistors to warm the car have been known to go out, mine did and had to have it replaced but usually it doesn't still run, it just dies all together when you try to put the heat on! FYI.
I would say that your ignition switch is not working correctly, because that is what supplies the fan blower motor with the voltage and current to do the job. when you shut off the key and take it out, there should be no more electrical power 'what so ever' going to the heater blower motor.