The Constitution does not allow someone who does not meet the eligibility requirements for President to legally serve no matter what the election results are. It's right there in the Constitution under the Twentieth Amendment, Section Three:
"3. If, at the time fixed for the beginning of the term of the President, the President elect shall have died, the Vice President elect shall become President. If a President shall not have been chosen before the time fixed for the beginning of his term, or if the President elect shall have failed to qualify, then the Vice President elect shall act as President until a President shall have qualified; and the Congress may by law provide for the case wherein neither a President elect nor a Vice President elect shall have qualified, declaring who shall then act as President, or the manner in which one who is to act shall be selected, and such person shall act accordingly until a President or Vice President shall have qualified."
A few notes.
1. There is no such position as a "President elect", legally, until such a time as Congress has accepted the results of the electoral college votes and a person is actually named as the "President elect". This means that the term "shall have qualified" refers to something other than the results of winning an election. There is only one place left in the Constitution having to do with "qualifications" for the office of President, that being the eligibility requirements from Article two.
"No person except a natural born Citizen, or a Citizen of the United States, at the time of the Adoption of this Constitution, shall be eligible to the Office of President; neither shall any Person be eligible to that Office who shall not have attained to the Age of thirty-five Years, and been fourteen Years a Resident within the United States."
2. Since it is the duty of Congress to name an interim President in the event of a President elect's "failure to qualify", they, Congress, must know whether or not to do so. This means that they, Congress, must be aware of whether or not a President elect meets the eligibility requirements from Article two. It is the burden of the President elect to "qualify" or "fail to qualify", thus NOT proving one is eligible under Article Two to Congress is the same thing as "failing to qualify." Congress avoiding its duty to uphold Section 3 results in the same "failure to qualify", something they may have done on purpose in this instance depending upon the reasons why. National security? Who knows at this point.
3. How was Obama's eligibility proven to Congress without a valid long form birth certificate? He apparently does not possess such a thing or we would have seen it a million times by now.
4. The eligibility requirements start out with two simple words which forever preclude anyone who "fails to qualify" from serving as a legal president, "No person". Someone who sneaks in because Congress failed to uphold it's responsibility to enforce the Twentieth Amendment, Section 3 doesn't legally exist. The Constitution cannot be fooled just because Congress didn't act when it was supposed to. A President elect either qualifies or he cannot ever be President, period.
Thus it is that we have protection from someone who is ineligible to serve as President already written into the Constitution. Unfortunately, we also have a Congress that did not uphold it's oath to support the Constitution and a usurpation of the office of President is the result. We know he is illegal strictly on the basis that we don't know if he is eligible. If he "qualified", there would be no debating the subject. The fact that nobody in Congress is able to say whether or not he is eligible means that he never proved to them that he was and thus has "failed to qualify".
Now, as to who has "standing". Any elected official at the state or federal level who took the oath of office in Article Six has standing, to demand that the Constitution be obeyed. This means that no judge can deny them the enforcement of their oath to "support the Constitution" if they have a question about whether or not any portion of that Constitution has not been adhered to. In this case, the Twentieth Amendment, Section 3 has clearly been IGNORED by the primary party instructed to act under it, Congress.
If Congress can essentially ignore and thus "nullify" a provision of the Constitution, what is law? What good is a system of laws that those at the top of the food chain can ignore while suppressing those of us near the bottom through illegitimate enforcement. They derive their powers from the very document that they are ignoring. This is a farce and they must be called on to admit it by facing the truth. Nullification. If they can do it, so can We the People.
The Constitution does not say who is responsible for checking the candidate's credentials. It would take legislation passed by Congress to do that. And I'm not aware of any such law being on the books. Maybe there should be, but currently there isn't. So Obama's qualifications or lack there of fall into a grey area. Maybe he is qualified. Maybe he isn't. But he isn't required to provide any more than he has, and there is nobody tasked with looking into it.
There has been no finding by any authorized body, judicial or legislative that Barack Obama failed to qualfy under Article 20. He stopped being President-Elect when he took the Oath of Office on Inauguration Day.
A federal judge ruled on this in 2009 in a lawsuit that went all the way to the Supreme Court of the United States:
Barnett v Obama, US District Court Judge David O. Carter: “There may very well be a legitimate role for the judiciary to interpret whether the natural born citizen requirement has been satisfied in the case of a presidential candidate who has not already won the election and taken office. However, on the day that President Obama took the presidential oath and was sworn in, he became President of the United States. Any removal of him from the presidency must be accomplished through the Constitution’s mechanisms for the removal of a President, either through impeachment or the succession process set forth in the Twenty-Fifth Amendment. Plaintiffs attempt to subvert this grant of power to Congress by convincing the Court that it should disregard the constitutional procedures in place for the removal of a sitting president. The process for removal of a sitting president—REMOVAL FOR ANY REASON—is within the province of Congress, not the courts.”—U.S. District Court for the Central District of California, October 29, 2009
http://ia600204.us.archive.org/1/items/gov.uscourts.cacd.435591/gov.uscourts.cacd.435591.89.0.pdf