The problem with this question is that the Bible is not a single book, and as such does not directly address what "other books" are considered canon.
Peter, for instance, writes that Paul's books are highly profitable for reading, "even if they are sometimes hard to understand," but there is no book anywhere that says "these books are canon, these are not."
Indeed, the "closed canon" of the NT wasn't even recorded in its entirety until A.D. 367, when Athanasius published his festal letter. This letter is merely a recommendation list, much like the NY Times best-seller list. It lists books which are "profitable" (see 2 Tim 3:16) for a Christian to read, but it merely reflects consensus, not dogma.
As such, there has only been common consensus as to what is canonical and what is not. (This is also why it is downright silly for people to talk about "suppressed" books or "hidden gospels", because frankly there is no authority on what is or is not canonical.)
That said, for the NT, one of the "marks" of canonicity has been aposotolic authorship. While scholars will tell you that Paul, for example, probably didn't write many of the "Pauline" letters (and never claimed to write Hebrews!), they were traditionally ascribed to apostles. (This is why Jude got in - apostolic claim). Because the original 12 apostles are no longer writing, this avenue of new canon is pretty much closed.
One biblical admonition that is often cited is Rev 22:19,
and if anyone takes away from the words of the book of this prophecy, God will take away his part from the tree of life and from the holy city, which are written in this book.
This can't really be used for canonicity claims, however, since "this book" would only refer to Revelation itself - a completely separate work from the other 65.