The last time I testified in court, I was asked to swear an oath to tell the truth, the whole truth, and nothing but the truth. The final part of the oath, "so help me God" was omitted.
Not being a bigger fool than my mother gave birth to, I did not make an issue of what struck me as a rather serious lacuna at the time. The problem, however, is, just who was I asking to attest to the truth of what I was saying? An oath using the traditional formula calls God (or, in ancient times, a god or goddess) as a personal witness attesting to the truth of what is being said, with a very strong "or else" hanging over the head of the perjurer, even if the State doesn't catch him or her at it.
As administered, the oath I took merely relied on my personal truthfulness - of which the court had no knowledge, official or unofficial. This is a serious problem, for an oath is a way of committing yourself to something in an extraordinary fashion, a calling in of a deity to punish you if you speak falsely and for taking his, her, or its name in vain. Without that, the oath is, essentially, meaningless. By stating something in a court of law (or on your tax return, or other official document) you are held liable for telling the truth. Those little lines that contain words to the effect that you are attesting to the truth of what you're saying under penalty of perjury may be a handy reminder, but they are not an oath. You're guilty of perjury even if you refuse to take the oath, or scratch out the notice about perjury. An oath is a formal, additional assurance that you are telling the truth, on the assumption that no sane person would risk eternal damnation for a transitory good obtained by lying.
The problem in a religiously pluralistic society such as our own, however, is that some groups may not recognize the Bible as a holy book, or recognize any book in that category, for that matter. That being the case, someone taking an oath - that is, someone who is committing him- or herself in an extraordinary manner - should be allowed to dispense with using the Bible and use a holy book or something denoting the sacred in his or her religion (or lack thereof) ... but the decision should be on a case-by-case basis, not made for us by the State.
Frankly, most non-Christians when faced with a situation such as swearing on the Bible are able to make a "mental reservation," and justify it on the grounds that, while the Being (or beings) they worship does not recognize the special status of the Bible, that whom they worship will accept the oath as being made in his, her, or its name. The only exception to this would be a form of oath that implies a belief in a god other than their own - which the traditional form of oath in a court of law does not; it did not specify which god someone was swearing by. The Bible was a symbol - an important symbol, but still a symbol. It is not necessary to administer a binding oath, but it is useful as a visual reminder of what you're actually doing.
So, yes, the Bible should continue to be used for swearing-in ceremonies and in courtrooms - with reasonable provision made for people who feel uncomfortable with or do not believe in the Christian revelation.