My first thought is most modern TVs have a "game mode" setting which reduces input lag and makes for a more optimal playing experience. The Xbox and TV "communicate" to enable this setting when the Xbox is turned on.
I had a really nice Samsung that would sometimes do weird things when switching over to game mode. Just software glitches.
I would possibly look and see if you can disable some of the automatic settings on the TV. Basically tell it to NOT auto detect gaming consoles, etc. You should be able to just manually turn on game mode when you want it. Or, sometimes with some TVs, you can tell it to always keep a certain HDMI port assigned to game mode so that it doesn't try and switch back and forth.