I know that Mormons say the Bible and the Book of Mormon don't contradict. I don't know about that, but I would like to know whether there actually is a verse in the Bible that could support the belief in an exterior source such as the Book of Mormon, so highly regarded -by Mormons- as another revelation from God.
The problem is that the Bible doesn't really have an infallible table of contents. So, while there is nothing in the Bible definitively suggesting that there is another set of infallible documents inspired by God which relate to the Old Testament, the Bible also doesn't exclude that possibility either.
Because I have seen multiple comments on this, I think it might be good to address the two verses in the Bible which condemn addition or subtraction. The first is
Now, in the context, it is fairly clear that the "Word" is the Tanach (the book of the Law, which is the first five books of the Old Testament1). The verse preceding is talking about the law, and the verse after is talking about when the law was written and what has happened to those who have strayed.
If this is to be taken in the sense that it is referring to the whole Bible, this means that we must take that to exclude the New Testament. After all, Christ most certainly added to and subtracted from the Law (the dietary restrictions are removed, while the laws on forgiveness and loving of neighbor were expanded). So, clearly, this must be taken as a reference to the book of Deuteronomy itself. Especially since there were prophetic and historical books written both before and after Deuteronomy2.
Ok, so we will posit that the canon of Scripture is the sole deposit of faith (a point I am not willing to cede, but I will allow us to make that assumption for the sake of argument). But, what is that canon? The early Church could not agree for one hundred years after that phrase was written.3 So, if the people who knew the author had no idea which books he was referring to, then it is more than presumptuous to assume that we know. Since it is clear that we are missing books from Paul4, can we really assume that something else isn't lost? After all, the Pauline Epistles are scripture5. Of course, then there are questions of the Deuterocanon (you might call it apocrypha), which I simply do not have the time to deal with at the moment.
Then there is the question (assuming that we can agree on what should be in the canon) of which version. As a Catholic, I can definitively state that the woman caught in adultery is part of scripture6, but can everyone else? The unfortunate answer is, "No, they can't7".
So, if we are to assume that Rev. 22 is referring to the Bible as a whole, we have to admit that we aren't sure which documents it is referring to, or which books. That pretty much means that unless we can find a collection of documents which dates to the late first century, we can only asume that some piece of the NT has been added or taken away. This means that your copy of the Bible is probably blasphemous and should be thrown out.
Then there is the problem of translation. In order to translate scripture, you have to make choices. Some words simply don't translate well. Others are ambiguous. My favorite example is Gen. 3:15
The problem is "He shall strike your head." In more modern manuscripts that is clearly הוּא, and you will notice the little dot by the wav. Unfortunately, that dot was added in the middle ages, which means that the word, "he" in that sentence could very well be the ancient word for "she".
So... what are you left with? The Bible you have at home is a dubious collection of dubious manuscripts collected and translated by a dubious translator. Considering that this is the best you will ever have available without knowledge of at least several versions Hebrew, Aramaic, Syrian, Greek, and Latin8 (and possibly French, German, and Ecclesiastical Latin, and even then you are probably not qualified9), are you really willing to say that your copy is necessarily infallible? Are you really willing to say that you have an inspired table of contents? Honestly, I hope not. Because if you are, then you are basically saying that the canon of scripture fell out of the sky... That is a story which sound oddly familiar.
1. Testament is actually an antiquated translation of the Greek word for covenant.
cwallenpoole's answer is credible: the Bible is not complete as we have it today, so we run into "negative evidence" fallacies to say that because it doesn't mention something, the something must be false. (That would be like saying "Jesus Christ" of the New Testament is false because that Name is never mentioned in the Old Testament.)
Before I also attempt to answer your question, let me prelude with a disclaimer: most members of the LDS church will hesitate to show "proof" or even "evidence" of the Book of Mormon's truthfulness, for the very existence of the Book itself is evidence of what is known as the Restoration, which is the fulfillment of prophecies concerning the dispensation of the fulness of times (the days before the Second Coming in which the Lord would send prophets to the earth again after a falling away first).
Mormons will remind anyone that the only way to truly believe in the Book of Mormon, or have a testimony of its truthfulness, is not by scholarly study, but by reading it and praying to ask God if it is true, and that an answer will come by the Holy Ghost.
So with that in mind, the following verses (in addition to those I've linked to) should be instructive. Keep in mind that while many people interpret these verses in many ways, this is one way the LDS population understands them:
There are others. This list is not comprehensive, but I hope others will add to this as is proper.
The Bible and the Book of Mormon support each other. The Book of Mormon supports what the Bible says. All the prophets since the world began (including those in the Book of Mormon) knew there would be an apostasy of the Christian church in the New Testament and that God would set his hand a second time to recover His people. For example, see 2 Nephi, chapters: 27, 28, 29, 30, 31, 32, and 33, and Mormon 7:8-10 are a few, although the doctrine of Christ's mercy and Atonement is found all throughout the entire book.
Of particular interest is 1 Nephi 13 (especially the second half of the chapter) which discusses Nephi's vision of the settling of colonial America, the coming of Christians from Europe, and the introduction of the Bible ("which proceedeth forth from the mouth of a Jew") among the Native Americans, that it is holy and true, but that many plain and precious parts are actually missing.
So, I add these references to the Book of Mormon because they should help support the Bible references mentioned here that help to directly answer your question.
Finally, the notion of "infallibility" in scripture is misleading, because while God does inspire true scripture, imperfect but holy men write it down.
From the Title Page of the Book of Mormon (emphasis added):
Matt briefly mentioned Ezekiel 37:15-23 in his answer, but it deserves greater emphasis. The linked article starts out by very briefly mentioning writing on staves, and then discounts it, saying that had Ezekiel meant a staff, he would have used a different word. And then it goes and gives several examples of 'ets meaning all sorts of things, many of which had their own specific words. But on the other hand, we have many translators and commentators throughout the ages calling Ezekiel's "wood" a staff, a rod, or a stick. People centuries closer to the author (and thus more likely to be familiar with his cultural context) than we are consistently call the "wood" a rod of some sort, so it's worth understanding why someone might do what Ezekiel describes to an actual stick.
LDS scholar Hugh Nibley has a thorough treatment of the subject in his article, The Stick of Judah. He begins from the premise that what Ezekiel is doing, writing on two halves of a rod and joining them together so that they become one in his hand, is familiar to his audience, even though it's not familiar to us. Then he explains the historical context:
Since ancient times, formal contracts have been represented by inscribing the terms on a rod and then cutting it down the middle, through the inscription. One half was given to each party to the contract. When it came time to fulfill the contract, both parties would bring their sticks before a priest, magistrate or other legal authority, who would join them together in his hand. This made fraud impossible, as only if the two pieces of wood matched perfectly, grain for grain, could the contract be verified, and changing the engraving would require physical access to both halves.
This use of tally-sticks to signify contracts is of ancient date, and survived until surprisingly late in history. The official British archive of fulfilled tally-stick contracts was kept in the Houses of Parliament until 1834. (Interestingly enough, the official written language of such contracts was not English or even Latin, but Hebrew!)
So the Lord has Ezekiel prepare a tally-stick, signifying a formal contract in which one side represents writings of Judah, and the other, writings of Joseph, that will be joined together into one "in my hand." The writings of Judah are obviously the Bible, which leaves an obvious question: what can the identity of the stick of Joseph be? This prophecy requires a writing similar in nature to the Bible, containing the Lord's covenant with his people, as the Bible does, but coming forth from Joseph, not Judah.
The Book of Mormon contains the sacred records of an offshoot of the House of Israel descended from the tribes of Ephraim and Manasseh, the sons of Joseph. It speaks of God's commandments and his gospel, and testifies of Christ just as the Bible does, and it specifically mentions that its words are to be joined with the testimony of the Bible. (2 Nephi 29:8, read the entire chapter for context.
In other words, the fulfillment of Ezekiel's prophecy requires something that looks exactly like the Book of Mormon, and for which no candidate other than the Book of Mormon exists that I know of.
A little different Mormon angle is what you would read in Amos 3:7.
According to that verse the Lord doesn't do anything without consulting his prophets. This makes perfect sense because that's exactly what the bible is, the Lord gave his prophets direction in how to lead his people and the prophets wrote down it down.
Following this logic, it makes sense that wherever God has had people who followed him he would give them leadership through prophets, and those prophets would write down the direction they were given in the form of scripture.