Regarding the messages being on the TV, do you mean while something else is on? If so, you're going to need hardware that injects text into the video stream (which may be an expensive thing to set up). Interaction with those devices will be another issue in itself... If you don't care about anything else being on, displaying text with a Pi is trivial.
As for the iPhone connectivity, the simplest solution would be an app that reads the messages and connects to a server on the Pi over wifi and sends the messages through.