Macworld summarizes findings first reported by Engadget. Apparently a bug in the iPhone clock app prevents non-recurring alarms from properly triggering on New Years day.
I was able to confirm this after a couple of false starts. For the bug to show itself, your iOS device must actually tick over from 11:59 p.m. on December 31, 2010 to 12:00 a.m. on January 1, 2011.
The work around for now is to set up the alarm as a recurring event.9to5Mac claims that the problem corrects itself after January 3rd. In the meanwhile, be aware if you use your iPhone as your alarm clock.
Update: Apple has officially acknowledged the problem:
Apple spokesperson Natalie Harrison confirmed to Macworld that Apple is aware of the problem. “We’re aware of an issue related to non repeating alarms set for January 1 or 2,” Harrison said. “Customers can set recurring alarms for those dates and all alarms will work properly beginning January 3.”