I used to, back when I had a full size server rack in my living room.
Gaming experience? Nearly the same, as my 55" tv at the equivalent viewing distance of my 24" monitor looks pretty much the same. Of course I was sitting on the floor with my keyboard and mouse on the coffee table, but otherwise very similar. Mouse and Keyboard didn't matter, I just ran a very long usb extension cable to a powered hub on the table, and plugged all my peripherals into that.
Upsides?
-Gaming using a controller. Sit back on the couch and have fun. Just like a console, but prettier.
-5.1 audio. Sure my TEACs are badass, but they are 2 channel and I am not settling for some crap logitechs. With the PC feeding through the Onkyo, my neighbors knew full well that I needed additional pylons.
-Picture quality. My TV currently stomps my monitors in picture quality and contrast ratio, plus I could tinker with things like 3D and 240fps barf inducing interpolation.
Downsides?
-My back. Sitting on the floor with a non adjustable height coffee table makes for some uncomfortable sessions.
-Input lag. If you are lucky your tv will only cost you a frame or two, other tvs can cost you 5-15 frames, have fun with that.
-Singular activities. Wife wants to watch a movie? No pc for you.
-My porn. Unless you live by yourself, it's a little difficult to hide 55 inches of vagina on your wall. Pictures on the walls and any other reflective items become spy tools.
-Actual work. Most people don't have separate gaming and regular computers, so the first downside is expanded upon when you have to do non fun stuff.