I do have an inverter that I can run the television off of, but it's not quite like you're thinking. It's not a built in inverter, but rather a little portable inverter that is plugged into the lighter plug where the television is, and I have to move the power cord from the TV to choose either the inverter or the regular 120V wiring of the RV.
If you plug the television into an outlet that is powered by the inverter, it should work off of inverter power. While driving, you won't have much luck receiving broadcast television, but you can watch DVDs or play video games or whatever over the television. (Digital TV broadcast signals do not work well with moving receivers.)