Fuel consuming units use LESS fuel while stationary, but they still use some. That's why I tend to station a fuel-using unit at a currently unused airfield, so the fuel-using unit draws a steady supply of fuel and supplies to the airfield in case I need to use it quickly. That is ideal for bombing Partisans out of supply with a TAC, then the fuel-using unit rolls out to accept their surrender. As an example, one TAC can cover all of occupied FRA, with the aid of a handful of either CAV+AC divisions or single MOT brigades at a few key airfields. The TAC flies to the closest airfield, and can begin bombing immediately due to the presence of fuel and supply.
No idea whether or not fighting uses more fuel/supply than moving.