It is the actual recording mode that generates the heat. Transcoding, putting stuff in the buffer, saving to storage, etc. With the HDMI-out, you are just sharing the viewfinder and offloading all those tasks to your computer, so heat is not really an issue unless your room is extremely hot to begin with.

The sony a5100 is really popular with twitch streamers right now because it will run all day in this mode and you can turn off all of the UI elements to get a 'clean' output. You don't get all the cool cinema features but you do get a camera that is better than just about any webcam.