I would think that would be the job of the Supreme Court. They are the ones charged with swearing in the President. Since they are charged with upholding the Constitution, they would be obligated to ensure that the person they are swearing in is qualified for the office.
Since this issue has been brought to their attention, I think they are obligated to rule on it before the electoral college votes. I think it would be a dereliction of the constitutional duties to refuse to consider it. They may be free to ignore the evidence and rule any way they please, but I do believe they have an obligation to consider the issue.