I believe I've eliminated IR weirdness from the equation. I took the PC, my desktop monitor, and the keyboard/mouse into the room with the TV. First I powered on the PC and monitor (exactly as when on my desktop), and the remote control/Flirc worked perfectly. Next I powered on the TV and played with the TV remote. The PC/Flirc/remote continued to worked fine. This implies to me that the issue has something to do with the HDMI connection between the TV and the PC.
Just to be clear, there was not a connection of any kind between the PC and the TV in this test. I wanted to see if there was IR coming from the TV perhaps.