Yes, I can confirm that when media identified as a "TV Show" is highlighted/selected, there is a "Cover Art" -> "Get from Internet" menu item, and that it does indeed get cover art for at least some TV shows, but I don't know from where.
When it finds cover art, the first image thumbnail in the list of thumbnails is identified as, for example, "Your Library - 680x100 (276KB)", while the following thumbnails are just identified as, for example, "680x100 (276KB)". Some TV programs have lots of cover art images, so they can't just be images on my hard disk. They must be coming from the internet.
"Get from Internet" works fine for music as well.
There is no "Get from Internet" item in the "Cover Art" menu when a movie is selected, and if "Get from Internet" is in the list of recently used functions when a movie is highlighted, it does nothing. There is no attempt to lookup any cover art. If I change the "Media Sub Type" for a movie to "TV Show", then "Cover Art" -> "Get from Internet" shows up in the menu again, and the function attempts to look up cover art, but finds nothing. When I said it was working in my first post above I must have had a TV show highlighted, or a movie which was marked as a TV show. Sorry for the confusion.
As Matt says, use 'Get Movie & TV Info' for movies, and if you want to use something other than the single image that is returned, manually find and load the cover art you want.