My view would be that when you have ignition on all the ECUs are consuming power and much of it is wasted as heat hence a fan will come on consuming even more energy.
OK, next to actually travelling it's not very much. Nonetheless, that consumed energy will need to be replaced by a higher charging current when your engine is running. So it simply costs you fuel that you might otherwise not burn.
I don't have a hybrid but I imagine when you move off you are just using the traction motor. I cannot imagine that what you are doing is reducing stress on the motor and therefore extending its life.