I could be wrong but dry mode could mean "dehumidify" mode.

HOW TO: Use an Air Conditioner in Japan | Surviving in Japan: (without much Japanese)

If you use the dehumidify function, you can save some electricity versus using the air conditioning function, although it, of course, won't really cool things down (usually heats up the room instead), but just reduces moisture in the air, which can feel better in some cases. You might also see this function as ドライ.