I have the same problem. Error -6722. When I went to firewall options I figured out that it was TURNED OFF.
And even in this case I was facing the error -6722.
I'm trying to mirror my display using Apple TV (3rd G) and MB Air (2012 version) on Mountain Lion Air Play. So the devices are not the problem too.
The fact is that when I do mirroring with my Bluetooth turned OFF, it all works very nice.
When I turn it on, seconds after, the error occurs. So, it seems to me that bluetooth is causing it.
But, "why should I mirror it on my TV if a can't use my keyboard and magic mouse"?
My two questions are. 1. How can I fix it? 2. Does someone know if I can have my Mountain Lion payment back (because I bought it just to mirror and use the TV as display) ?
If someone is able to help me I would be very grateful.