The short answer is: yes, it does, although in an indirect way, never a direct one.
The long answer is: provided that the ambient's air is well above the freezing point of water, that the cabin's air filter is clean and that the system is set to cool ambient air and not recirculated one, the evaporator's thermal load is proportional to fan speed, and, therefore, air volume.
If a larger mass of warm air has to pass through it, more energy is transferred, through the evaporator's fins (so that even the evaporator's design and, in particular, its exchange surface play an important part) from the air to the liquid refrigerant allowed inside it by the TEV or orifice tube so it expands more and, along with the absolute pressure inside the evaporator, the refrigerant's vapor superheat (the delta between the boiling point of the fluid at a certain absolute pressure and the temperature of the vapour) increases, since after expanding into saturated vapour, it has enough time to catch enough heat to warm up further by vaporizing the remaining liquid (an important property of a superheated vapour is that no fluid in the liquid state is carried around by the vapour, unlike with saturated vapour).
If the system is provided with an OT, well, it's pretty straight from here: a hotter vapour at a higher pressure reaches the compressor, and each piston inside each cylinder, in turn, has to pump that vapour through a small discharge valve on the head's reed plate: that is, the head pressure increases, the opposite force experienced by the piston during its movement increases and this results in more engine load.
With a TEV system, the spring inside the TEV allows a certain superheat value to be maintained, and the sensing bulb in contact with the evaporator's outlet raises the valve's needle allowing more liquid refrigerant inside the evaporator when the outlet becomes warm and less when it becomes cold, all of this resulting in a vapour with a certain superheat and pressure at the evaporator's exit and therefore in the suction line.
If a smaller mass of air has to pass through the evaporator, less energy ends up in the refrigerant liquid, so the evaporator cools down faster. The TEV needle begins closing down more and more allowing less and less liquid inside while still letting a constant stream of superheated vapour get out of the evaporator, while, with the OT, any liquid slug that made it past the evaporator without turning into superheated vapour collects in an accumulator before reaching the suction port of the compressor. The lower suction pressure translates into a lower head pressure and a lower engine load.
Turning air recirculation on makes the evaporator cool down to the required temperature even faster and easier since all it has to cool is already cooled air. Therefore the compressor's head pressure and engine load decrease even more when the recirculation is on.
The minimum load on the engine is even lower if the compressor is able to adjust its displacement to the actual evaporator's temperature (with internally controlled variable displacement compressors) or, even more efficiently, to the HVAC system's cooling demand (with externally controlled variable displacement compressors) up to the point that its pistons are travelling a very small stroke when, in the former case, the evaporator's temperature is near the water's freezing point or, in the latter, when the evaporator's temperature is at the temperature required to cool the air coming out of the vents down to the desired temperature.
By a similar reasoning, since the cooling capacity of the system is influenced also by the airflow over the evaporator at a present time (along by the flow of liquid refrigerant through it and its exchange surface), the cabin's temperature is influenced by the fan speed. A larger airflow means more cool air that can cool the cabin further. A smaller airflow means that the vents give a smaller flow of cool air that, although being able to let the evaporator cool more so that the vent temperature actually goes down, other heat sources (such as the sun's rays whenever the glasses aren't tinted) can easily restore the heat taken away by the evaporator and keep the cabin temperature higher than when the airflow is increased.
All of these reasonings (plus a not strictly related one, that is the amount of heat provided by the heater core) are automatically taken into account by ATC (automatic temperature control) systems whenever you set the temperature you want into your cabin. With manual climate control, you have to precisely know how your car's HVAC system works to use it to maximum efficiency.