It's when you have a heated argument with another person, and he/she keeps telling you 'oh my God', by implying that they don't want to hear you speak, is when you use the Don't "Oh My God" me, which means you are right, and the person should listen whether he/she likes it or not.
Jennifer: Stop talking and listen to me. He is mine, and you should leave him alone.
Candy: Oh my God, when will you shut up about this?
Jennifer: Don't "Oh My God" me.... you know I'm right about this. He loves me, not you.