How to turn off your TV receiver with Apple TV remote?

If your AV receiver supports HDMI-CEC including commands for standby from other devices, and all HDMI devices in the setup have CEC activated, then the AV receiver should go to standby when the Apple TV box goes to sleep. And reversely wake up with it. Manufacturers can each determine if and which CEC commands are supported; no function is mandatory, except HDMI passthrough of the CEC signal itself.

Apple TV box: Settings﹥Remotes and Devices﹥Control TVs and Receivers﹥On

TV: See its manual. The TV manufacturer may have its own trade name for CEC.

AV receiver: See its manual. It may be permanently on (passively: reacting to other devices).