Which one is grammatically correct?

There are bigger issues with this sentence to worry about. “When a group of people goes out to dinner, if I put my card down, you don’t leave the table…” Switching from “a group of people,” which is abstract and third person, to first person (“I”),to second person (“you”), it feels sort of like being thrown into a whirlwind. I don’t know why you’d start this sentence with “when a group of people goes out to dinner,” because that makes it sound like a hypothetical group, not when *you* go out to dinner with a group.

Both equally work 

"...until you Venmo me" would be my choice, but both are correct: "...until (after) you've Venmo'd me". What I find more awkward about this sentence is the use of both "a group of people" and "you/me" -- swapping between 3rd person and "me" and the generic "you" isn't very natural here. If you were talking to a friend, it would be better to say, "When a group of us go out to dinner..." or maybe "When I go out to dinner with you (all)..." If you're speaking generically (like in a Reddit comment), you'd be more likely to say, "When I go out to dinner with a group..." and then the generic "you" is more explicit.

Both work

Both, depending on the context. But the sentence as a whole is awkward. You should change “a group of people goes” to “we go”. It’s not “a group of people”; it’s the speaker and the listener (at least). We use “we” for that. It sounds unnatural and unusual phrased this way. I also personally wouldn’t say “if I put my card down”, but rather “if I pay”, though some people would say this, I’m sure. Also, “you don’t leave” sounds really demanding and kind of rude. That’s sort of how you would discipline your child, not how you would talk to your friends. It sounds like you’re angry at them. I would say: If we go out to dinner and I pay, please Venmo me before you leave. This sounds much more natural and wouldn’t come across as potentially rude.