Official line: Don't hot swap.
Advice to original question: you can do it to get where you need to go, but don't make a habit of it.
I have seen many, many cars come in without a battery at all. We hook a jumper up, start it, remove the jumper and drive it in. Never hurts anything.
That's not to say it can't, but I've seen it done many times before and it hasn't hurt anything.
By and large, you'll be better off not doing it, but in a pinch, it will not hurt anything.
Think about it. The car is running off the alternator, not the battery. If you disconnect the battery while the engine is running, you haven't introduced a load to the alternator. You've actually reduced the load, especially if you just started the car. I've seen cables removed from running cars many, many, many times, more than anyone else that has posted in this thread, and regardless of what they SAY can happen, we've never seen it.
Again, to be safe, don't do it. The kid in question seemed to be in a bind, and given his situation, he could easily do it and not have a problem.
Edit: Skoorb: yes, the negative is always the first to be disconnected, and last to be connected.