George Washington's inaugural Bible has been on display a number of times at the National Archives, and George H.W. Bush used it in his inauguration. Son George W. wanted to use the historic Bible for his inauguration, as well, but he was stymied by bad weather. Washington's Bible stayed inside in the Archives that wintry day.
Office holders don't even have to swear to uphold the Constitution. The Constitution itself gives them the option of affirming their support, in order to accommodate people, such as Quakers, who believe the Bible prohibits the swearing of oaths.
Presidents Franklin Pierce and Herbert Hoover, who was a Quaker, affirmed their oaths instead of swearing them. Hoover used a Bible for his affirmation, as did Richard Nixon, also a Quaker.
Nixon, in fact, swore on two Bibles, but as UCLA's Volokh points out, "this didn't seem to help."
In the courtroom, the Federal Rules of Evidence require witnesses to declare, by oath or affirmation, that they will testify truthfully. But that's not always as simple as it sounds, either.
Convicted 9/11 conspirator Zacarias Moussaoui was asked to stand and take an oath during his trial, but he refused and showed utter contempt for the judge. He said he stood by his Islamic oath and claimed he was telling the truth—-a claim ended up being about as truthful as much of his testimony.