Barack Obama was not sworn into office on the Quran, this is a false statement that is part of a range of speculative statements regarding Barack Obama's religion. The statement actually references another politician, Keith Ellison who is a congressman from Minnesota, who is the first Muslim member of Congress.
Keith Ellison became House representative for Minnesota in November of 2006 and announced that he would use the Quran instead of the Bible during the Congressional ceremonies. Certain members of the American public raised an issue regarding this, showing that this statement operated on the mistaken belief that a Bible or religious text is used in the ceremony at all. The Congressional swearing-in ceremony is done in mass with incoming representatives swearing an oath to uphold the Constitution.