IR-video is possible with FPS override and/or display gain. So far, I've only used it for timelapse (f/5,6 320 ISO and 1FPS in normal daylight with a 50D and 720nm filter), but the same exposure translates to roughly f/2, 1000 ISO and 30FPS, so regular speed IR-video is possible. Pushing the ISO above 25600 may be necessary for indoor-situations, even with relatively good lighting. A compact camera or camcorder with a built-in night mode will probably give you less noise.
For filming in darkness (to the human eye, anyway), you'll still need a strong IR light source, though. I've seen something like it with several strong video lights, each with an IR filter in front of it.
A true "IR-mode" is not possible without physically replacing the IR-cut filter with a piece of clear glass. The best way to do it with an out-of-the-box DSLR is to get a 60Da which has slightly better IR-sensitivity (around 690nm maximum, I think)