Well, guess what?
I can't show what this looks like because it doesn't show up well in photos, but there are two very fine vertical lines running down the entire height of the screen: one line running equidistant from each of the left and right sides of the image. They are exactly where the borders of the "pillars" are when you watch a standard (not high-def) image without "stretching" it to fill the screen. Since most of the interesting programming around here is standard definition and I don't like distorted images, the tendency is to watch television "pillarboxed". I guess that's how the lines got in there: with the borders always being "on".
It's supposed to not be a permanent condition of LCD screens. And technically it's not "burn-in" at all, in the sense that plasma screens can get it. It's properly called "image retention"... but that doesn't keep it from being any less a nuisance. Right now I've got the TV on with the image stretched to fill the screen, so that the pixels making up those lines will get "exercised" and start working right again. I'm hoping this will work. If not, looks like that service plan is going to turn out to have been a great investment after all.
In the meantime, if anyone's ever had any experience with this kind of an issue, I'd sure appreciate some advice. I like to try to fix things "in-house" as much as possible, without having to lug something as big as an HDTV back to the store :-)