When you mean "Latency", you mean high ping on servers or very long delay responses from your input / output devices? If it's something related to ping you should check your internet connection since it has nothing to do with TV or your Monitor, now if you are experiencing input lag or response delay from your TV you should check your TVs settings regarding video / smooth tecnologies and Refresh Rate.
An advice, I use TV for gaming as well but don't "duplicate" your monitor, instead you extended option to extend your main screen to another monitor or Tv so you won't suffer from performance loss, also by doing that you can decide in which of one of those you wan't to keep as your main desktop screen, you can set either your TV to keep...