Zip Your Lip can have two meanings depending on the context. In both, the image means to keep your mouth closed (as if it were zipped).
In the first, it means to keep a secret. The person talking to you is asking or instructing you not to tell anyone else. Can you keep your lip zipped about it? It could be a secret, rumors, or gossip. Those are always good to zip your lip about, don't you think?
The second is just a request (or command) to keep quiet in general. If the person is extremely frustrated, they might use a shorter form. Zip It! ;)
An alternative that means the same thing is Button Your Lip (or Button It!).
Question, comments, suggestions ... please don't hesitate to let me know. Until next week! :)