I'm having a similar problem with my iPhone 4S. I rented a movie yesterday (on my iPhone) for my plane flight. I watched most of it, but I'm home now and I want to watch the rest of it on my computer because it's a lot more comfortable than watching on the phone screen. I've read all the support documents, done a lot of google searches. The support document says to select the movie and "click move." But there is no "move" to click when the movie is selected in iTunes. If I right-click, I can rate the movie or get info on it, but that's it. I can't even play the movie in iTunes when the iPhone is selected. I have the latest version of both iTunes and iOS. Since I started playing yesterday, I have only a few hours left to finish watching the movie.
This is the first and probably the last time I'll rent something through iTunes. It's been a disappointing experience from beginning to end.