Of course it burns extra gas. Where do you think the energy comes from??

ANY electrical load you put on the car will make you burn more gas. Even turning on the radio will make you burn more gas.

If we assume you're drawing 11 amps at 13.2 V, and if we assume the alternator is working at 20% efficiency, then according to my calculations you're robbing the engine of about 1 horsepower. That’s not much, but it's certainly not zero, either.

But it doesn't matter... none of these schemes work.