If Obama falsely and knowingly swore in an official document that he was born in Hawaii, he could perhaps be impeached and convicted on the basis of perjury. There is no evidence of which I am aware of his having taken an oath to that effect; if such evidence exists, and should it be proved that he was not born in Hawaii, perjury would probably be an impeachable offense.
In any event, for impeachment and conviction of "high crimes and misdemeanors," on the basis of perjury or (improbably) without a criminal statute, it would be necessary for President Obama to have had some clear knowledge that he was not a "natural born Citizen" when he assumed office. Where might evidence of such knowledge be found?
If Candidate Obama actually believed that he had not been born in the United States, but knowingly claimed falsely that he had been, evidence of that would certainly be relevant to his state of mind. Proving what someone "believes" is never an easy task, unless the believer has written or spoken of his beliefs. President Obama has probably never expressed, publicly at least, a belief that he is not a "natural born" citizen. It has been claimed (probably correctly) that President Obama's Hawaiian birth certificate is not probative of his birth in Hawaii, even though copies of it were posted on his campaign web site. It seems reasonable to presume that Candidate Obama was aware of the posting. Whether he believed that the Hawaiian birth certificate was spurious is a different question, and a big one.
If President Obama, as a youth, traveled on a passport issued by a foreign country, that could be probative that he was then a citizen of another country. There are two possibilities here:
1. Obama was a United States citizen but obtained and used a foreign passport. If he did not thereby intend to renounce United States citizenship, then he did not forfeit it.
2. Obama was not then in fact a United States citizen, and therefore used a foreign passport. This would be very good evidence that he knew that he was not constitutionally qualified as "natural born" when he sought and accepted the presidency.
There are many questions and few answers. In any event, it seems very unlikely that the present House of Representatives would impeach President Obama even if a newly discovered and clearly valid Kenyan birth certificate proved beyond question that he was born in Kenya; it also seems unlikely that the present Senate would convict. In practice, an impeachable offense boils down to what the House and the Senate say is it is. Former President Ford said substantially the same thing. Currently, the Democrats control both houses of the Congress, and President Obama seems to control the Democrats — if not completely, then still sufficiently to ward off a successful impeachment. The very first impeachment and conviction in United States history of a (Democratic) president seem so unlikely to occur while there is a Democratic Party majority that the subject may not even be worth discussing.