I think this is normal. I blink a lot when playing osu!
You may not notice it, but there are periods where you don't blink at all for up to a minute. Either caused by you finally starting to play harder maps that don't give you time to blink, or just concentrating harder unnecessarily. That's when your eyes get irritated because the surface of your eye is not getting the needed lubrication, yet still moving around a lot. That's why you feel the need to blink incessantly thereafter. It doesn't happen in other games because you don't move your eyes that much in other games, and a lot of the time in FPS games where you don't blink, you're just staring at one spot.
Personally I hate using a fan or having anything even remotely related to a breeze in my face. My eyes dry out almost instantly when playing osu!, but if I'm not playing osu!, I can probably stand using a fan for about 3 hours without a problem, although only from the side. It's because you're not blinking at first, and then blinking a lot afterwards, that you think there's a problem.