I don't know about Nikon camera abilities in regards to the question asked, but on my Canon 5DMKII and 7D cameras, everything said above is incorrect. I just tried it with both. I removed the lens and was able to shoot stills and video no problem.
Without the lens on, I did get a text on the back screen of the 7D saying to make sure there was a lens attached and to press stop/go to proceed. But then, even without attaching a lens, I pressed stop/go, and started recording, no problem. It didn't try to stop me any further. The 5DMKII had no warning at all, and just went ahead and started recording.
The results were the same. No focus, obviously, but I got a recording out of each, and as I moved the camera around, the colors changed to reflect what it was aimed at. Here's one of them:
ETA: I figured it would work before I even tested it, btw, because there are plenty of lenses being used with adapters, bellows, extension tubes, etc., that don't communicate with the lens, so I couldn't see how the camera would know the difference between whether one of those was being used, or if there was no lens on the camera at all. Testing confirmed my thoughts - at least on the Canon cameras I own.