I recently got a 55" LM7600. Haven't play a lot of games on it, but I do experience more lag when I play Battlefield 3 online multiplayer, when I have the 2D->3D feature on. The control is way too sticky to aim correctly, like a split second delay on you aim/movement. Playing it in 2D, there is much less aiming lag, but still a little (but can't tell if its just the game or the TV, as BF3 always have a little bit aim lag a few months back, and its been a while since I play it again). Probably won't affect single player games much.
In terms of the 240Hz, I don't think it has much to do with gaming lag, but it makes things move smoother. I like watching movies with that smooth look. If the game lag, it still lags regardless of 120 or 240.