A Christian who believes in the Bible shouldn't swear on one.
The Bible, of course, is worse than useless when it comes to consistent advice on morality. But the New Testament (to avoid confusion, ignore the Old Testament on this one) is pretty clear about one thing: Christians shouldn't swear. Not to God and not on the Bible or on anything else.
Here's what Jesus (supposedly) said about it:
But I say unto you, swear not at all; neither by heaven for it is God's throne; Nor by the earth; for it is his footstool: neither by Jerusalem; for it is the city of the great King. Neither shalt thou swear by thy head, because thou canst not make one hair white or black. But let your communication be, Yea, yea; Nay, nay: for whatsoever is more than these cometh of evil. Matthew 5:34-37
And the Book of James condemns swearing "above all things." I guess that'd make it the worst possible sin.
But above all things, my brethren, swear not, neither by heaven, neither by the earth, neither by any other oath: but let your yea be yea; and your nay, nay; lest ye fall into condemnation. James 5:12
But what about Muslims? Is there anything in the Quran that forbids them from swearing on it? Well, not that I know of, but let me know if there is and I'll note it in the SAQ.
Of course this shouldn't be an issue. The swearing-in ceremony doesn't require the use of the Bible, Quran, or any other religious book. So Representative Elect Keith Ellison shouldn't have a problem getting sworn-in, although he should leave his Quran at home.
But what about Christian representatives? Since the Bible forbids them ("above all else") to take any type of oath, they can't take the oath of office without renouncing their faith.
Oh well, that shouldn't be a problem for them. Most are just pretending to believe for political purposes, and the rest are mostly just hypocrites.