Typically only when in recording mode. If the video is pass through you should be fine. You will however likely need a dummy battery you can plug in.

It won't be recording video to its SD card, just outputting the moving image you get in the little preview screen out of HDMI as well.

I guess the heat comes from the expensive encoding, which doesn't happen here. But yeah, the camera should ideally also be able to powered by cable and not just battery only. And it should have a setting so that the preview screen have no UI elements...