"Climate control" is airconditioning. Apparently on a car, the air is chilled, and then run through the heater matrix to warm it to the right temperature. So the answer must be "yes" as you are chilling the air first.
A car starts up when it is cold, and has an optimum operating temperature.
If the car is under its optimum operating temperature, then having the heater on (assuming the car sends hot water through the heater matrix before it is up to temperature, and I have no idea whether that is universally true) must increase consumption.
If the car is at or above operating temperature, then it has to get rid of its waste heat somehow. Better that it is put into the cabin than that that the engine fan has to be turned on (which will use a small amount of electricity (= petrol)).
Equally, of course, the heater fan will use a modicum of electricity (= petrol).