1) First Amendment establishment clause: "Congress shall make no law respecting an establishment of religion"
2) No explicit religious qualification listed in Article II, Section 1 under Presidential qualifications for office:
"No person except a natural born Citizen, or a Citizen of the United States, at the time of the Adoption of this Constitution, shall be eligible to the Office of President; neither shall any Person be eligible to that Office who shall not have attained to the Age of thirty-five Years, and been fourteen Years a Resident within the United States."
3) Article VI prohibits a religious test for office:
"but no religious Test shall ever be required as a Qualification to any Office or public Trust under the United States."
4) Article II, Section 1 under the section describing this subject allows EITHER an oath OR an AFFIRMATION:
Before he enter on the Execution of his Office, he shall take the following Oath or Affirmation:
"I do solemnly swear (or affirm) that I will faithfully execute the Office of President of the United States, and will to the best of my Ability, preserve, protect and defend the Constitution of the United States."
No mention of a Bible is made here - there is no such requirement and in fact such a requirement would contradict Article VI and the First Amendment and an affirmation is just as valid as an oath.
Finally there is no requirement that swearing must be done upon a Bible. One may swear without one.
http://www.law.cornell.edu/constitution/constitution.overview.html
