Be silent. A demand to not discuss a subject. One should not discuss a taboo subject. ‘Zip it’ refers to closing a plastic bag or zipping up a jacket, so in this sense it is to close one’s mouth and no longer talk about something. (Compare to Put a sock in it, Shut one’s trap or Shut up.)

“When my son spoke to me yet again about buying a car , I reminded him has a bad driving record so he should just zip it and leave the subject alone.”