My alarm didn't go off this morning! I set an alarm up and I can see the alarm symbol at the top in the notification bar. However, the alarm didn't go off this morning! Thankfully I'm still in the crossover where my old phone is still in use.

Does the phone have to remain switched on at all times in order for alarms to function?

It's a Galaxy S running 2.1 still, because apparently we're still waiting for Orange to decide how much bloat they want to pack 2.2 with before release.
Yes, the phone has to remain switched on for the Alarm to work.