According to the Constitution, the courts don't get to remove presidents. Only the congress can do that. Even if the birthers really came up with the goods, the Supreme Court would merely cite Article I and toss the case over to Nancy and Harry.
True, but someone who is not eligible to be President, isn't President. It would be a logical contradiction. He can be de facto President, he be acting as President, but he can't be President. He can only be a usurper.