I think I may have discovered the problem. As I was replacing the HDMI cables and double-checking my connections, I realized that the TV port I have my receiver running into (HDMI 2) is labeled HDMI/DVI. I assumed both HDMI ports are the same, but apparently the Yamaha receiver doesn't like any ports that are also designed for DVI.
I moved my receiver to port 1, which is HDMI only, and moved my Tivo mini to port 2, the HDMI/DVI port. So far I've had no more red screen.
My Tivo mini is always on so I anticipate no handshake issues with it.
I hope that was my problem. It could just be the re-seating of the cables that fixed it, though, or just luck so far.