In my setup I can't afford to not be able to use the tuner, so I'm stuck. I wouldn't be surprised if running a signal through the encoded circuitry is getting it warm while not having an input gives it less to do.
I did try a recording to see if even though the video output was locking up if it would still record. Oh boy! I was in SP/2 hour mode which normally gives excellent recordings. The machine hung up within 10 minutes and from the very first parts of the recording the video was horrible. Huge blocks on the screen. It was as though the encoder was trying to compress the data so much that you could put it on an old LP record
You might want to try running your unit with an input for a couple days, even not recording, and see if you run into any problems.
Hope you have better luck!