A television that has high refresh rates doesn't automatically mean it's going to have high input lag. If I'm correct (and please correct me if I'm wrong), they can achieve those refresh rates through certain image smoothing techniques and options, which you can enable, configure and disable at your leisure. On my Sony, that option is called Motionflow. When enabling it, it does increase input lag, but for most games I haven't had an issue with it. I've been playing Ni No Kuni (a Japanese RPG) and Tomb Raider (a third-person shooter) and the various Motionflow options haven't impacted my performance in either game. In something like Call of Duty or other competitive first-person shooters, though, it may be best to disable such features.
It's hard for me not to use the image smoothing effects in games, however. The interpolation technique makes movies look strange (the Sony has non-interpolation options, as well), but in games the effect is absolutely fantastic. It basically does what those 120Hz monitors do - and does it better - but without having to reach 120 frames per second to see a smoother image.