Does a car burn more gas when the heater is on?

Yes, a little, but much of the heat is generated by operating the car.

Heat is virtually free. The only thing you pay for is the fan and for most cars that is on all the time anyway.

A/C costs you. The full up compressor load can be 2–3 hp. Let’s say you drive a small car and your A/C draws 0.5 hp. If you are cruising down the road at 60 mph you might have a road load of 15 hp. That means you will be using 0.5/15 or about 3% of your fuel to run the compressor. If yo are in city traffic the percentage is greater since road load drops.
You can get the numbers for your car by going to fueleconomy.gov . There is a wealth of info there.

Yes. I don’t know exactly how much more, but it isn’t a huge amount more, I think the heat comes off the engine, so it is not actually producing extra heat, but air conditioning the system actually has to work hard to cool things down, so air conditioning uses more extra gas than the heater.

A car will burn more gas with the AC turned on but not that much. If you open the window the extra drag that causes will burn more gas than the AC. So I wouldn’t worry about it.

No, the car heater uses minimal energy and is free.

Your air conditioner is the gasoline energy culprit in the summer.

Yes, it does. Most hypermilers do not turn on the heat, they wear parkas instead.

Here are some great winter hypermiling tips

Personally, we use the heat, but on our larger vehicle (not our Prius) we use a grill block. See above, 2nd link

No, the heat is relatively free. It is waste heat from the engine. If you don’t use it it gets wasted anyway. There may be a cost to running the fan but that is minimal.

Air conditioning costs a little because the engine has to run the compressor but remember, the compressor doesn’t run all of the time unless you are in Texas in the summertime in a black car that has been sitting in the sun.

No, because a traditional engine has to get rid of heat anyway.

It’s different for turning on the lights or radio.

Turning on the heater is a problem for electric cars.