My previous CPU case was left off all the time, under constant "fanning."
One day, my CPU wouldn't boot. The problem? Something had shredded up my motherboard. Whatever it was, it couldn't have happened had I kept the case closed, which would have kept my computer fairly secure from outside threats. You'd have to be raving mad to keep your case open all the time, in my humble opinion.
Even if that isn't the cause for you, I'd reccomend, if you can, examining the card itself. Since you keep the case off all the time, it should be fairly easy to remove the graphics card, examine it, and plug it back in. I've never had to do this, so I'm not sure if you could even recognize any damage, however.
If all else fails, take your CPU to a bona fide computer repair shop.