I have a 720p set.

All of a sudden after a year of satisfactory use, I have problems with 1080i. I have my STB set to display both 1080i and 720p, but now, when I switch from a SD channel to a 1080i channel, I get sound but no picture. Going back and forth again corrects the issue, as does unplugging the HDMI cable and replugging.

I've forced the STB to only do 720p, and have no issue now.

Is it the tv, or the STB that might be at fault, or even the HDMI cable itself?

You don't mention the make/model of your TV, but it's probably a handshake issue between the TV and the STB. Many (older) Sonys have had these issues and setting the STB to 1080i Fixed usually solves the issue. See the following post on the topic of HDMI issues:

http://www.digitalhome.ca/forum/showthread.php?t=71112

I believe you may have a Sharp TV based on some earlier posts. Perhaps your HDMI cable is not making good electrical contact at both ends... Unplugging/replugging both ends to ensure a good contact may help, or it may not. Setting the STB at 1080i Fixed or 720P Fixed will, very likely solve the issue since you will no longer be changing formats.