I would differ and say that holding two passports would and should be challenged if such a person ran for President. Obviously, it would be a political liability—at least for now.
The U.S. Government does not recognize dual citizenship. Any person who currently holds two passports is ineligible to serve in the Office of the President, Office of the Vice President, and any other Federal or State elected office.
Undivided allegiance is a prerequisite to holding an elected office and taking the oath to protect and defend the Constitution of the United States of America.
I would say it might make a difference whether a natural born citizen eligible under Irish law to obtain an Irish passport actually does so or not. And I agree it would be eyebrow raising and possibly a political liability if he did. I don't think it would be disqualifying. However, if the person never exercises the privilege, then his Irish passport eligibility would merely be an interesting biographical detail.
Before 2008, I didn't think we'd ever elect an anti-American to the highest office in the land. We might as well have elected Billy Ayers, who is natural born beyond question and a better writer, to boot.