Ok, that does sound like your TV may be sensing a No Signal condition even when you do have a signal. A lot of newer TV sets actually sense if the TV is actually getting a signal, some expect you to push buttons on your remote so the TV knows if you are actually watching TV, and others sense people moving around the room, indicating that people in the room are not sleeping in front of the TV set....
Those are all energy saver settings on the newer energy saver TV sets. I think they've been around since at least 2009, and I've had customers complain about their TVs doing that to them.
Here is how you check that --
Press your menu button, arrow down to the SETUP menu, then right arrow into it. Now arrow down to your ENERGY SAVING setting, arrow right into it then select OFF. If it is off, it won't turn off if your unit monitors the activity of the remote control or a proximity sensor that monitors that you are moving around in the room. It may also deactivate the No Signal shutdown.
That may stop the TV from at least turning itself off.
Right now, I'm just throwing out the "possibilities" that I've seen on TVs. We'll try to take this a step at a time, but I would also suggest you check your Plug and Play mode -- That is also listed in the SETUP menu settings. I suggest STARTING with the Plug & Play settings when the TV's input is set to TV, not Video 1 or HDMI 1 etc....
You do the Plug and Play first, simply because the ENERGY saver settings may get set back to a Low/Medium/High setting. That's why you do that AFTER Plug and Play so it remembers you setting it to OFF.