Wow, I guess I just had no idea cooling was important in a home theater setup. I did take care to make sure components aren't stacked right on top of each other and have adequate airflow where they vent, and my receiver has nothing on top of it at all. But I had no idea that the products were made such that you have to add cooling to keep them from melting down over time. That's pretty sad.
It turns out this is the second failure in as many weeks for my home theater setup. My DLP bulb died last week, and I had just gotten the replacement in about two hours before this new issue with the receiver occurred. I don't know how the two could possibly be related, but it is strange that they happened so closely in time.
Right now, I think I'm just going to continue to use the receiver as purely an audio device hooked up to my TV's optical output. It's not ideal, but I don't want to throw any more money at this right now if I don't have to. I can probably solve the hassles with switching inputs by getting a better universal remote that will send out macros with discrete input selection codes to the TV and receiver. It might actually be better this way in the long run, as I'll now have more inputs to play with on the receiver for audio devices.