When the Mac no longer gets a signal from the projector/monitor it will 'turn off' its video port. In this case the Mac is losing the signal from the projector when the AV Receiver is either turned off or switched to a different device to the Mac mini.
What you need to do is trick the Mac mini in to thinking the projector is always present, it will then not deactive the video port and will not reset the video settings. Back in the days of VGA connections you could make a custom plug which hardwired one of the pins to a signal to trick the computer however these days when using DVI or HDMI connections you need to get a purpose made device. The one I have used along with many other people is the Gefen HDMI Detective Plus (or if your using DVI still the equivalent which is the DVI Detective Plus).
See http://www.gefen.com/kvm/dproduct.jsp?prod_id=8005
or http://www.gefen.com/kvm/dproduct.jsp?prod_id=4715
With these the Gefen device learns the signal from the projector or monitor, thereafter even if the projector or monitor (or AV Receiver in the middle) is disconnected or turned off the Gefen device keeps sending the signal so the Mac thinks it is still connected and keeps the same video settings active.
