So... when you perform the Onkyo's built-in calibration or test tone, does everything come out the speakers it says its testing?

What audio format is the Onkyo receiving (like, what it says its receiving, not what the TV says its sending)? Is it Dolby Digital or DTS? If so, the decoding should be done inside the Onkyo which should make what you're experiencing impossible.

What sources is this happening from? Just TV apps sending audio over ARC? Do you have any other devices hooked up, and if so, how?

If the TV is sending LPCM, there could be some kind of incompatibility, ie. its sending a 7.1 signal and your receiver is trying to decode it as 5.1 or something.

Does the TX-NR616 support eARC? Do you have enhanced HDMI support on or off on the TV?