I've never had an issue with heat from the TV, and it's been on for hours and hours as it's used as the screen for playing media and music as well. There is no cooling fan installed.

There's no chimney within the wall. If you're referring to the fireplace, it's a direct vent gas fireplace that vents straight out the back of the exterior wall.

Here's a little more info about the surrounding cabinets on also. The cabinet to the left is also setup for equipment if needed depending on how much equipment you have. It was also pre-drilled on the left for a small fan install (it was never needed though) which is released through the top of the cabinet next to it.