I'm heading the door/car/barbershop.
This version isn't correct. As a commenter said, you can't head a door. Note that head is a verb, though. You can head a soccer ball for instance.
I'm heading for the door/car/barbershop.
This version is correct. It has slightly more urgency than the example below. For example, you might say a soccer player is heading for the ball, because they are in a hurry to get there.
I'm heading to the door/car/barbershop.
This version is correct. It implies a longer distance than the other versions. It makes more sense to use this version if your car is parked 3 blocks away, as opposed to in your garage, for example. However, it works fine in all scenarios.
I'm heading towards the door/car/barbershop.
This version is correct. You shouldn't use this version, however, unless you are currently in transit. For example, if you were in your house, planning to go to the barbershop, you wouldn't say you're heading towards the barbershop. But if you're in your car and someone asks you where you're going, it would make sense to say you're heading towards the barbershop.