News reaches us direct from an unnamed Apple Genius that there’s an occasional bug with the Apple TV (4th Gen) operating system update routine that means the update is not fully applied. As a result the Apple TV will refuse to turn on.
The solution is simply to unplug it from the power for around an hour. This resets the system management controller (SMC), allowing the unit to come back to life.
This is definitely worth trying before you take the unit into an Apple Store, or mail it back for repair/replacement.