I had both a DLP projector and an LCD projector (using one right now) and am looking into a new projector right now, since mine is dying (in case you want to know why: I got an Epson 5910 LCD, which is great, but Epson decided to make it a bit more silent and didn't cool it enough, so the filter wheel is dying... and that's not really repairable. Damn them!)
With my old DLP projector, I had terrible rainbow effect issues, but that was 8 years ago and it was a 3-segment 1-Chip DLP projector.
Looking through the details of all the tech, I found out the following:
Picture quality:
LCD doesn't have rainbow effects. The black usually is not as good as with DLPs, but honestly, I found the picture of mine great.
Unless you buy a 3-Chip DLP, you will have some rainbow effect (whether it's visible depends on yourself).
DLP Projectors for around 500 EUR will certainly have a visible rainbow effect, for about 1000 EUR and higher, you'll get 1-Chip DLP projectors with fast 6-segment wheels which will minimize the rainbow effect (making it invisible for most users). WIth even more money, you get even more segments and at some time 3-Chip DLPs.
DLPs are a bit sharper than LCDs, but I never really had an issue with the sharpness on mine
The brighter the picture on a DLP is, the more visible the rainbow effect will be, so you can imagine how a cheap one with high brightness will be!
3D:
If you want to watch 3D, LCDs have a ghosting issue. Mine also has that. My wife doesn't see that, but I can clearly see it. DLPs are supposed to be better here.
Be sure to get a 144Hz capable one, in case you want flicker-free 3D pics.
Gaming:
According to what I found, DLP projectors usually are a tad faster for gaming than LCDs (but whatever you get, be sure you can disable all picture enhancements, as that will slow down things for sure! It's often called gaming mode).
However, it always depends on the games you play. Most modern games are being built with some delay in mind, so they play fine.
I had no issues playing stuff like Sonic Generations or Duck Tales Remake on my projector, but playing old games like Sonic 2 or the original Duck Tales was simply impossible
Other tidbits:
From what I read, LCD needs a bit more cooling so is a bit louder than DLP.
My current favs I'm looking into is the Epson TW6600 (LCD), the BenQ W1110 and the Optoma HD144x