I’m not sure that’s the question. The president is a citizen like anyone else and is subject to the courts like other citizens. If Obama were being ordered to appear in his capacity as president, a separation of powers argument could be made, but the question of eligibility to run is not related to his current status as president, and would apply equally to a private person running for their first term. If such a person would be subject to the court’s subpoena power, why should Obama escape it by virtue of the question being as to his eligibility for a second term rather than a first?
For what it’s worth, I don’t buy the various birther arguments, but I also wonder how, in a putative situation where a non-citizen did try to run, we’d ever get to the bottom of it. Candidates are not above the law, and people ought to have a mechanism to challenge eligibility.
Obama should have responded, in some fashion, perhaps through legal counsel.