Not necessarily curse, but the term does mean that what you are saying or doing is, or shortly will be, making me angry. If I do open my mouth to say something about the situation, bad things will be said. It is a warning to the other person to "start acting right!". Mother to a young child. "I see you are not eating the food that I gave you. Don't make me open my mouth." Boss to a subordinate. "The report is not yet on my desk. Don't make me open my mouth." Wife to a husband . "You want to spend that much on a television??? Don't make me open my mouth."

The situation I imagine it being said is one in which the speaker is afraid of embarrassing themselves. "I'm meeting the head of the department. I'm so nervous! Do me a favor at the meeting and don't let me open my mouth."