Projectors are one of those things that seem neat at first for gaming until you really look into it and find out all the flaws with it. For one, response times on all but the best projectors (read: $900+) are terrible. A $100 1080p monitor would have a better response time than a $1000 projector. Two, you have to have the perfect environment as well as the perfect display in order to get the best out of your projector. If you don't have the perfect lighting with the perfect wall size with the perfect, blank wall (or projector screen), you're not going to get the best looking picture. Three, you'll have to either already have an audio setup or purchase one because, like most TVs, projector speakers (if it has one) are utter ass. Like, $20 speakers probably sound better than a projectors. Then there's the amount of power they take just to run, some of these run at
250W-300W. That's like 3 or 4 40" TVs. They're also loud cuz of the cooling they require, but that sort of varies by projector so meh
At the price point you have to be at in order to truly have at least a "decent" projector experience, you might as well just buy a big ol' 1080p TV, or hell even a 4K TV would be cheaper.
I mean, it's really neat and all (I used to have a projector setup in my garage that we'd use at family reunions for games and stuff) but it's simply not...feasible for long-term gaming.