I was unable to find any information whether WWII tanks (especially those used in North Africa) had air conditioning or not. Likely there was some kind of a ventilation to get rid of the fumes, but how effective it was in lowering the temperature inside? Are there any known cases of tank crews fainting from the heat?
Desert Warfare: German experiences in World War II by Major General Toppe mentions briefly that the crews just had to take it. With the hatches closed. What's worse, sometimes what ventilation system they had had to be shut down to reduce fuel consumption in combat!
Tank or no, it also mentions that it was customary to give 3 hour rest period at noon whenever possible.
Related question: How warm was a tank? Would crew sleep in a tank during a cold winter night?
There was a lethal cocktail of vapours in a T-34 during battle.
We once had a loader and then a driver pass out during battle, which sent us into panic as the tank was designed to advance when the throttle pedal was raised, not when it was lowered. We opened the hatch to let the worst of the foul air out and within a few moments the driver was aware enough to apply the brakes. We were desperate to get out into the fresh air, but we could hear the rounds splatting against the armour and knew we would be dead in seconds... Being in a tank in July 1943 was like being placed in a hot often pumped full of toxins and suffocated.
Although from Kursk, and not the desert, the above passage gives a good indication of how desperately poor conditions could be inside a tank. This being the era before smoke extractors were fitted to tank guns, whenever a round was fired and the breach opened, the fighting compartment would fill with cordite fumes to add to the smells of oil and fuel and unwashed bodies of the crew.
Further info indicates that the dry heat of the desert was not such a problem as might be imagined
“…even in tanks and armoured cars the desert heat led to no disastrous incapacitation. The high evaporative power of the desert air saved the heat-regulating mechanism of the body from any undue strain, so that as long as the tank engine, and hence the engine ventilating system was pumping air through the vehicles conditions were not dangerous. In the desert therefore there was no call for special air-conditioning devices for AFV’s.”
This would seem to be confirmed by the general lack of commentary on the matter in books dealing with the desert war.