Apple says Do Not Disturb will fix itself on January 7, until then you're on your own
Apple has released a knowledge base article that says iOS 6's Do Not Disturb scheduling feature, which for some reason decided to take the New Year off, will start working again on January 7. Apple also recommend that, until then, anyone experiencing the issue turn Do Not Disturb on and off manually.
Apple doesn't explain what caused the issue to begin with, or whether or not they'll fix it in the next software update so it doesn't happen again in the future. (Though presumably they will issue a fix, given how often time-based problems have plagued iOS, a little reassurance that Apple takes it seriously would be nice.)
Have you experienced problems with Do Not Disturb staying off, or refusing to turn off, since New Years? If so, is Apple's response enough to mollify you?