I have had no heating problems whatsoever. Mostly its been in use for 1-2 hour sessions at a time, occasionally longer. I do turn the camera off in between though, so it's not turned on 24/7.
Most annoying is a bug in the Canon webcam driver, if the camera is switched off, it displays a static image, which somehow fully occupies a single core on my machine. I would have thought encoding a still image into a video stream should be doable with less cpu cycles.
This is a confusion of terms. By "recording" people can mean both writing data to the card and activating the sensor to receive information. The latter is what overheats a camera, and the process is the same for both sending data directly to HDMI or recording it to a card. Some older sensors on Canon digital SLRs would actually begin to burn themselves out just by having the screen display what the sensor was receiving (also known as "live view") for more than about 10 minutes.