Sure, but blaming air conditioning for sore throats is like blaming condoms for pregnancies; the connection between the device and the outcome is indeed present, but to ascribe immediate causation is logically flawed.
It boils down to this: if you don't use either apparatus correctly, undesired consequences can and do occur. The fact that these undesired consequences occur does not, in any way, detract from the fact that both of these inventions are superb and the majority of people who use them competently benefit greatly.
Some reasons air-conditioning could cause a sore throat:
- Poor installation leading to inadequate air circulation, causing the system to stagnate in certain areas and potentially brew infection. Solution: careful design and planning before installation, as well as installing a system that was designed to handle the air flow and quantity of the building into which it is being installed.
- Poor maintenance, leading to filthy air filters and potential system-wide problems such as mold. Solution: replace air filters on a regular basis, and have the system as a whole inspected/maintained at the time intervals recommended by the manufacturer.
- Always running the system with vent control closed, leading to inadequate air exchange between the home and the outside world, again potentially allowing air in certain environments to stagnate and possibly brew infection. Solution: compensate by occasionally opening the windows, or simply running the system with vent control open.
- Running the system to the point of excessive aridity, leading to dessication and irritation of the throat membranes as the dry air passes between the environment and the lungs. Solution: run the system slightly less, run a humidifier, or simply allow air exchange by opening the windows for a period. Staying well-hydrated also helps prevent this from occurring.
Every single cause above is an example of human error leading to use of an air conditioning system outside of its designed parameters. The fault is in how it is being used, not in the air conditioner itself. This is an important distinction.
Air conditioning is a modern marvel that has done wonders to lower our rates of heat stroke in summertime, particularly in the elderly; it is well-known that areas of lower socioeconomic status tend to suffer from greater heat stroke morbidity/mortality precisely because they do not have the resources to allocate towards air conditioning systems. Both the CDC and EPA have recommendations that advocate air conditioning use as appropriate:
 Of course, even if you DO use them correctly, there is a small baseline rate of failure.