I have a mid 2010 27' iMac that I use a lot, and last week my HD5750 GPU died : black screen, everything else working normally (I can access my iMac through shared screen with my MacBook Pro). Not a backlighting problem : no monitor on system preferences, "unknown GPU" on system report, Novabench can't run any graphics test.
I accepted that my GPU had died, and searched eBay to buy another one, but it's much too expensive (400$) for a 6 year old computer. I was about to declare my iMac dead (it just got "vintage" a few days ago), but I wanted to "see" what went wrong.
So I ordered ifixit repair tools and opened my iMac yesterday : so much dust, just unbelievable. I removed the graphics card, inspected it, cleaned it (I also cleaned everything I could access in my iMac). Then I put everything back in place.
BINGO !!! No black screen anymore, my graphics card works again. And my fans don't spin fast anymore, I can't even hear them.
Just to let know a "dead" GPU may only need some dusting.