240Hz TVs don't accept 240Hz signals. If the TV says 24Hz that's what it's receiving. Many movies are filmed at 24fps, and to display that properly you need a TV that can display 24fps or a multiple of that for displaying the same frame multiple times before switching to the next frame. W/ a 240Hz TV each frame can be displayed 10 times; 10:10 is another name for this. On a 60Hz TV 60 is not an even multiple of 24, so the 1st frame is displayed twice, and the next frame is displayed 3 times. This happens 12 times per second for a total of 60fps, and is called 3:2 pull-down. 3:2 pull-down can introduce some judder during panning shots, and is undesirable to most videophiles.
The 240Hz (and 480Hz, etc) TVs are somewhat of a gimmick. It can't receive a 240Hz source. Most sources will be 60Hz. The TV still refreshes the screen 240 times per second, so each frame just gets displayed 4 times. That is unless you have interpolation turned on (aka motion smoother, auto motion, etc). The TV will do some extra processing to add extra frames in the middle. This is also undesirable to most videophiles as it looks unnatural, and can add artifacts. At least it's adjustable on most TVs now, so that you can dial it in if you like it a little. For gaming it's best to turn it off as any extra processing can introduce lag.
But a 240Hz panel does have advantages over 120Hz and 60Hz panels in other ways. The faster refresh rate means the pixels can switch faster, so there should be less motion blur even w/ the interpolation turned off. 3D image quality will improve since a major issue w/ 3D in general is cross-talk; the faster refresh rates help to overcome this. And it's a newer panel, and will most likely have improvements in other areas that come w/ time.
I'm still figuring out the 3D stuff, but I'll add what I can. Someone correct me if I'm wrong on anything below.
The 3D TVs can accept a 120Hz signal for 3D input, and this allows 60fps for each eye. Gaming like this you'd ideally want to use vsync and have it locked at 120fps. If the card(s) can't keep up then the fps might drop to 90fps (if you are using triple-buffering), 60fps, or 30fps. 90fps and 60fps might still be acceptable in some cases as at 60fps each eye is getting 30fps which is OK for some games. At 30fps each eye only gets 15fps and that's a 3D slideshow. Three 480's should be fine, though.
You're making me feel old w/ that comment!