My projector actually has really low input lag times, and I have actually played csgo on it, quite decently, it really is great, I have never noticed any tearing or anything of the sort.
I don't get the rainbow effect with DLP's but just to be safe, I went with an EPSON 3LCD projector, my model is the EPSON, TW5210, a great projector, I am using a Blackwidow "grey" screen, it's uses a paint mix with Aluminium particles in it in a 4:1 ratio so the screen is grey but reflects much more light than a standard grey screen, thereby improving contrasts, with better blacks, and equally good whites.