Yes, and no.

1. Your camera will shut itself down or put itself in a "cool-down" mode in order to save the sensor from overheating. I've had my Canon T1i warn me, and then shut things down in order to cool off.
2. Could you develop hot pixels? Yes. (I would tend to the idea that any hot pixels are pixels that were prone to going hot anyway, and so you would have seen them sooner or later. For that matter, my T1i has several hot pixels, and hasn't been used as a video camera extensively.)
3. Hot Pixels are dealt with using several methods:
   • Pixel Remapping (I forget the procedure to do this or if the newer models do it automatically, but the idea is that the camera will remap your hot pixels and essentially ignore the data from them (averaging in the surrounding data). I suspect this may only work in JPG, though, as my RAW files will still have them occasionally. (Canon's RAW processor, though, did seem to take care of them, so it may just be Lr.) I am not sure if Video gets remapped.
   • Dark Frame Subtraction for long exposures (only for stills)
   • Worst case, you should have tools in your post processor that allows for taking care of those hot pixels in post.
4. Nearly every sensor has them, and if they don't yet, they will. Just a fact of life in the digital world.

Note that all of the above assumes you are running with Canon's standard firmware and not something like Magic Lantern. If you were doing that, then there would be a larger possibility of doing damage to the sensor if it were to permit the camera to stay on beyond the overheating warning threshold. But if you are an normal firmware, I see no reason to worry.