Can air conditioning cause dry skin?

No. Dry skin is not directly caused by air conditioning, but various other factors including illness, solvents and materials that are made to absorb moisture, including lime, some soaps which dissolve natural body oils, concrete dust, sawdust, etc. Air conditioning is designed to simply cool the surrounding air, not to dehumidify the air nor expel solvents or dust that will absorb moisture from the skin.