Can I turn on a Onkyo AVR with the Apple Remote?

You should get an on-screen menu for Advanced Setup (p.29) when you do:

1. ‘Receiver’ button on the remote, then ‘Setup’ button on the remote,
2. Repeat ‘Down’ until you reach “HDMI Setup”, then ‘Enter’,
3. Select “HDMI Ctrl (RIHD)”, then ‘Up’ or ‘Down’ to configure On or Off (p.35),
4. Press ‘Setup’ to close the menu,
5. Press ‘Return’ to return to the main menu.

Try to set “HDMI Ctrl (RIHD)” to On, see if that works,

else try “HDMI Ctrl (RIHD)” to Off, see if that works,

else it may not be programmed to take suggestions/instructions from other devices, as the function seems to be designed for the receiver as controlling device.

Let us know how that goes.

When you turn on the Apple TV box, it can then turn on other HDMI devices through CEC, if those devices support that.

It seems some or all Onkyo devices have CEC turned off by default (in favor of their own protocol). See: How to Control receiver with apple tv remote - Apple Community

For Apple TV, set: Settings﹥Remotes and Devices﹥Control TVs and Receivers﹥On.

If you mention the model, I can have a look at documentation, see if I spot anything on the subject.

You’re welcome :)