Nothing happens. I keep mine in a black case that generally hides the fact that its an iPhone. Its better that way
I've kept mine in a black leather case on my belt as well, so it's not as obvious that it's an iPhone unless I'm talking on it, or checking email or stuff.
On July 1nd, a friend was visiting from CA, and the first thing he did was grabbed the iPhone off my desk while I was looking for a place that sold banana leaves. Good thing he's been a close friend for 18 years. He's now gotten one of his own, but I had to demo it to about 20 of his inlaws when I went to a BBQ that he was hosting in Breckenridge.
At work, I needed to show someone a graphic off a web page at a meeting, and pulled out the iPhone to show it to him. Didn't have a moment to give him a demo, but at our staff meeting I had to give a 15 minute demo/talk about it as the word had made it's way though the company (it seemed) that I had one, and was asked as soon as I sat down at the table.
I was a little reluctant to let my two fellow gamer/anime geek friend/co-workers play with it, as they both have a "death to apple" mentality. I knew that they would never hurt it, although they wouldn't admit that it was cool, or really say much about it.
As for "in public"... I really haven't had the call to use it in public, although I guess I was using it walking out of the mall once, and out of a Borders another time. I really wasn't looking around to see if anyone was paying attention, and nobody really took my attention away from my web browsing at time, although safari crashed both times as I hit the parking lot!
-jt2