Does the Bible allow for additional (specifically Mormon) scripture? There are many arguments and counterarguments relating to this question, which has been a significant one in the mainstream Christian community for some time. (Note: for the purposes of this writeup I will exclude the LDS Church from the "mainstream Christian" community. While members of the Mormon church certainly consider themselves Christian the rest of the world happens to disagree vehemently.) There are two main points to consider when pondering this question:
Is there a Biblical-theological basis for scripture other than the Bible?
If there is, does the Bible specifically disallow any possible "additions"?
We will consider both of these questions in turn.
First, we must ask if there is any rationale (in the Bible itself especially) for other scripture? A simple analysis of the methods of God lends itself nicely to answering this question. Throughout the Bible, God reveals himself and His word to many different prophets in varying ways. (As noted in Hebrews 1: 1-2.) Each of these mortal messengers, or prophets, wrote about their experiences, and many of these works are included in the canonical Bible. (E.g. Moses, Daniel, Jeremiah, Isaiah, and Malachi.) However, the aforementioned Hebrews passage notes that God, "hath in these last days spoken unto us by his Son." Interpreted, this passage seems to state that Christ was the final word on the gospel. However, this interpretation is clearly false by the examination of the rest of the New Testament. While Jesus did not preach to the Gentiles during his mortal ministry, the Apostle Peter later received revelation (in a dream) that they were to preach to the Gentiles at that time. It seems clear that the keys of the Apostleship are the defining trait necessary for revelation and additional scripture after the time of Christ.
This then begs the question: Can there be additional scripture now that the original Apostles are gone? How can someone who did not see Christ's ministry be an Apostle? (Apostles are defined as special witnesses of Jesus Christ.) To answer this question we need only look at the case of the Apostle Paul. Paul was not a witness to the mortal ministry of Christ. In fact, he fought against the Christian Church vigorously. But on the road to Damascus he was visited by the Savior and this experience gave him the witness he needed to be a true Apostle. It thus is evident that Christ (through similar means) can call any man to be a true Apostle. It therefore follows that scripture beyond the Bible is certainly possible through the works of any Apostles.
Now we must consider the second question. Does the Bible specifically disallow any additional scripture? A few scriptures are quoted which purport to answer this question in the affirmative, but one is the most significant: Revelation 22: 18-19.
18. For I testify unto every man that heareth the words of the prophecy of this book, If any man shall add unto these things, God shall add unto him the plagues that are written in this book:
19. And if any man shall take away from the words of the book of this prophecy, God shall take away his part out of the book of life, and out of the holy city, and from the things which are written in this book.
It is important to note here that in the original Greek, the Greek word for scripture has been translated in the KJV as "book". Thus we see that John the Revelator has clearly indicated that no man is to add to nor remove from this scripture (i.e. the Bible.) But wait! When John wrote the Apocalypse, he was not ending the compilation of writings that would later become The Holy Bible. He was finishing his revelation and prophecy about the end of the world. For this scripture to refer to the entire Bible, John would have had to have looked into the future (not unlikely, as he was the Revelator), seen the Bible in its finished form (compiled over the course of a few centuries by men) and approved of its contents. This is not an unreasonable assertion, except that the original Catholic Bible contains more books than the current Protestant Bible! Thus all of the Protestant world, by this scripture, would fall under condemnation for taking away the words of his prophecy. In fact, for this scripture to have any validity as applied to the whole Bible, John would have had to specify one specific edition of the Bible published at a specific time, as they are all at least slightly different. Given that he did not do that, it seems much more likely that these verses refer specifically to the Revelation of John.
It is therefore clear that the Bible certainly allows for additional scripture, if that scripture is laid forth by Apostles (special witnesses of Jesus Christ). The Bible in no way disallows "additions" to its admittedly man-contrived canon. Therefore, the authenticity of Mormon (or any) additional scripture hinges on the Apostolic authority of its authors.