There is another thread that I believe talks about using the iPhone. But for what it is worth, I didn't have the courage to try this week's flight to JFK on Thursday. So I printed a paper boarding pass at home.
You can't save it to the iPhone that I know of. But you can click the link in your email and keep it on your screen from what I understand. I think the option I will employ if I try the service is to take a screen shot of the boardinb pass when it loads, and then open the picture when I need access to it. The iPhone takes great screen grabs.