No court has shot down any arguements. They only refused to here the cases for lack of standing.
1 - Birthers lack standing because they cannot convince anyone WITH standing to agree with them. Not a state official anywhere. Not a single member of Congress.
2 - Birthers have had cases heard in state courts, and they have been shot down. No federal court has then taken it up.
3 - Birther arguments were given the the US Supreme Court, and rejected without comment. That means either the Supreme Court found the issue trivial, or they considered it established law already.