not open (one's) mouth
To be completely silent; to remain discreet (about something); to not tell anyone (about something). All the while our teacher was shouting at us, we didn't open our mouths. Don't open your mouth about this! I would die of embarrassment if anyone were to find out.
open (one's) mouth
To say something, especially that which is indiscreet, boorish, or meant to be a secret. Why did you have to open your mouth about my crush on Tommy? Why couldn't you just keep it between us? You've got to make sure that he never opens his mouth about what went on here today.