It definitely can.. say you are applying to an average mid tier state school called X and you have listed that you have been accepted at 6 or so schools already some of which that are considered "better" than X state school, X state school might take someone over you because it looks like you most likely won't attend with so many acceptances already. Same thing if you have had 10 interviews already they might think the likelihood of you choosing their school is slim.
Obviously if you have amazing credentials and you are a perfect fit for them, they will very likely accept you, but when you are an average applicant or have border line stats (basically when the school has lots of applicants that look like yours or better and they have the choice to take someone else just like you very easily) it can definitely make a difference.
If it didn't make a difference AT ALL, I don't think so many people on MD App's would make such a serious effort to hide their stats or make their application so vague by not putting any extracurricular's down until the cycle is over.
It may not be a huge deal, but why put your application at any risks?