The rainbow effect is a visual artifact possible on many single-chip DLP projectors. Because there is only one image chip, color is created sequentially. In other words, all the red in an image is flashed on screen, than all the green, then all the blue. This is done fast enough that your brain combines them to seem like a full color image.
On older and less expensive single-chip projectors, this is done with a rotating color wheel. The sequential color action of slower color wheels can be seen by some people. This would be most noticeable with bright objects on a dark background, a streetlight say. Tiny movements in your eyes would make it seem like each bright object had a rainbow to it. Moving your eyes rapidly around the screen is another potential way to see "rainbows."
It is important to note that there are three groups of people when it comes to rainbows: those that can't see them, those that can, and those that can but don't care. There are some people who fixate on the rainbows, and can't think about anything else (the middle group). For these people, we recommend a three-chip DLP, LCOS or LCD projector instead.
If you can't see rainbows, DON'T LOOK FOR THEM. Why risk being part of group #2?
Most modern single-chip DLP projectors have fast enough color wheels that the rainbow effect is largely a thing of the past. Using LED lighting all but eliminates it.
For more on video projectors, click here.