I know that some LCD screens have issues when it comes to games on PC's. I wonder if the pixel refresh rates of LCD TV's are a problem with fast-action games or not.
LCD
response time was a problem on PC monitors until a few years ago. Response time and refresh rates are completely different. Refresh rates do not affect LCD displays, only CRT. A CRT must scan constantly to keep an image on the screen. An LCD displays a static image.
Anyway, I have never heard of response time issues on RPTV's or projectors, though I have only been keeping up with them for a couple years.
Interestingly, there have been lag problems with some DLP TV's. DLP has never suffered from response time issues and, like LCD's, is not affected by refresh rates. However, the electronic processing in some DLP TV's, most notably, Samsung, causes lag which is particularly noticable when playing games.
I would recommend an LCD for gaming. Front projection would be my top choice but only if you have a suitable light controlled room. The Sanyo PLV-Z3 and Panasonic AE700 are two excellent 720p projectors that would fit your budget and allow you to add an inexpensive screen.
If you want rear projection take a look at Panasonic, Toshiba, and Mitsubishi. LCD models.
Don't rear projection (and front projection) CRT's suffer from burn-in? I thought the burn-in risk on CRT projection was even higher than plasma.