To zip your lip means to promise not to tell anybody about what you have just been told.
1. The boss warned his staff to zip their lips about the launch of the new product so that other companies would not know beforehand about it.
2. The celebrity couple agreed to zip their lips about their debut film together and not divulge any details to the press.
3. Mahendra's friends zipped their lips about the party plans as it was to be a surprise for him.