"Sich einen Zacken aus der Krone brechen" is a very tongue in cheek way to say you consider somebody to be overly proud, pretentious, big headed, full of themselves - or, alternatively, simply freaking lazy.
The image refers, as Si Langbein already mentioned, to an actual crown. It's like a mocking "Oh, of course, we can't expect your majesty to help with the housework. You would run the risk of accidently breaking off a tip of your crown, or getting a smudge on your royal cloak!"
The phrase is mostly used negated as a request for the person to get over themselves and do something they don't want to do:
Du brichst dir keinen Zacken aus der Krone, wenn du auch mal den Müll rausbringst!
You won't hurt your status if you take out the garbage, too, every once in a while!
It's really not beneath you to take out the garbage, too, every once in a while!
The usage you quoted isn't typical, but more a play on words with the described phrase and the concept of Sternsinger, who often actually wear crowns with their costumes. So,
Das könnte manchem Sternsinger einen Zacken aus der Krone brechen: Von Königen steht in der Bibel nämlich nichts.
means something along the lines of
This could cause some Sternsinger to get butthurt: The bible doesn't mention a word about kings.