I've seen this idea baked in part, usually on the Sci-Fi channel, which will offer a chat-room dedicated to a particular episode or series of this or that, running only while the show is on.
What I have in mind for this idea, though, is a continuous service which offers not only running commentary
of the program you might be watching, but the availability of historical review of commentaries of shows that are over (more like a message board where you can go back and trace the progression of comments).
I envision access to this service both by a simple keyboard which is part of the television set, and also by web access where other users use a more robust messaging system on their computers alongside their television sets.
The service would provide for filtering so that you'd receive comments only from individuals whom you find interesting (which can be determined by popularity polls and other message board sections), and for some off the cuff content rating of television shows by users, which would have an actual 'hot-or-not' rating system for whatever show they were watching while they were watching it.
Access to the server would be free, supported by the web-site ads, and console sales for those who purchase the television attachment.
For the console, either there would be a two line black bar superimposed on the screen, top line for msg scroll, and bottom line for msg entry, or the console could have a small LCD screen to serve the same purpose.
The difference between this and the rest of the internet is that it would be dedicated, monitored for non-tv-related content (which would be removed), and also made as simple to access as possible to appeal to television watchers instead of people who draw more of their entertainment from television-independent web surfing.