The Torah makes reference to a Book of the Wars of God (Bamidbar 21:10-16). According to most opinions this was an actual written work which has been lost over time. It is not clear (at least to me) who the author of this work would have been. I would like to know what the status of this book would be, and why, should it be recovered. Would it be included in Tanakh (the biblical canon) and have a status similar to Divrei HaYamim, would it be treated as any other document from the ancient world, having no special status, or perhaps somewhere in between?
This is a very good question. I have tried searching for sources several times on this topic without much luck, but today I found a few sources that may assist in answering this question. Perhaps the general lack is because we know so little about this book and other books mentioned explicitly or hinted at in Tanach (such as Sefer Hayashar, 3000 parables of Shlomo, Divrei Shlomo1 and so forth) that it is difficult to draw any conclusions about these books. More sources that discuss these books are ambiguous about their halachic status. Tentatively I'll write down that from what I've seen, should these books be found B"H, their status would likely be somewhere in-between: Holy but not canon. Let me explain:
It's a mystery when exactly each of these books was lost, although I personally believe that no book mentioned in Tanach was still around by the late-Persian-early-Hellenistic periods, being that Ben Sira, which was likely written circa that time, mentions many of the leaders of Am Yisrael from Tanach but makes no mention of any person that does not appear our Tanach (save for the Kohen Gadol of his time, Shimon ben Yochanan, who may or may not have been Shimon Hatzaddik).
Furthermore, to my knowledge, Josephus, centuries later, also doesn't mention any people who might have appeared in such books, nor does he state that the canon, in his opinion, included more books. Some rabbis thought that in the case of at least some of the books, copies only existed in the Kingdom of Yisrael, and when the Ten Tribes were exiled, their books were lost as well. For example:
Rabbi Ahron Marcus wrote in Barzilai, pg. 275 (my translation):
"...this word, as well as the phrase "זרזיר מתנים", whose meaning is unknown, were most likely copied from sources that were brought from the destroyed Samaria, as it says in Mishlei 25:1: "These too are proverbs of Solomon, which the men of King Hezekiah of Judah copied"..."
Such an interpretation of the verse is hinted at in Tanna Devei Eliyahu Zuta, additions to Seder Eliyahu Zuta, Introduction 8.
Likewise, Pseudo-Rashi on Divrei Hayamim 1:7:13 wrote:
"The sons of Naphtali: Jahziel, etc. And why was no more of his genealogy traced? This is the reason, as explained at the end of Megillah Yerushalmi: Ezra found three books, and each one of them contained part of the genealogy; that which he found he did not write, and of the sons of Naphtali he did not find any more. For this reason, the entire genealogy is traced with omissions, because he skipped from one book to another and combined them. And that which he could not write in this Book, he wrote in the Book of Ezra. You should know [that this is true] because it is written close by (9:1): “And all Israel traced their genealogy, and behold they are written in the book of the kings of Israel, and Judah was exiled to Babylon because of its treachery.” The meaning is that if you wish to know the genealogy of the Ten Tribes, go to Halah and Habor, the Gozan River, and the cities of Media, because their annals were exiled with them, but as for Judah I found their records in Babylon, and that which I found, I wrote."
and so forth. Per sources such as these, it seems that these books were never outright rejected by Chazal as not being holy. It should be noted that within Chazalic sources, some traditions about these books may have been preserved. For example, Heschel opined that the phrase sung by Bnei Yisrael in the desert per Avodah Zara 24b and Beresheet Rabbah 54:4 was from the Book of Eldad and Medad (see a summary of his view here); Rashi on Megillah 14a thought that the words of the unnamed prophet in Divrei Hayamim 2:25:15-16 were those of Amotz (and see here), hence - likely that Amotz had his own prophetic work; and so forth.
Prophetic works not included in Tanach are, of course, holy - they are no less the words of Hashem than those that were included in the Tanach. The difficulty comes with regards to books that don't have names that sound like prophetic books - as is the case with the Book of the Wars of Hashem.
Today I found Chizkuni on Shemot 17:14 who opined that the book mentioned in the verse is this same book, and that it was lost due to our sins. This is pretty similar to what several other commentators wrote, but interestingly, one of the other books the he mentions is Midrash Ido, which was a prophetic work mentioned in Divrei Hayamim. He seems to equate these two in terms of status.
Based on this, I'd conclude that the status of the book is that of a lost prophetic work2, or at the very least, written in Ruach Hakodesh.
But were it to (hopefully!) be found, it would probably not be included in Tanach, as Rabbi Yehudah Hachassid wrote in Sefer Chassidim, siman 1016 (my translation):
"...and that which is written "are recorded in the story of the prophet Iddo" the Men of the Great Assembly have already said what was needed to be recorded and no prophet is allowed to add anything to the twenty-four books, not an angel or Eliyahu..."
I believe the reason for this is as the Talmud famously states (Megillah 14a):
"However, only a portion of the prophecies were recorded, because only prophecy that was needed for future generations was written down in the Bible for posterity, but that which was not needed, as it was not pertinent to later generations, was not written."
This is typically thought by many to mean that only these twenty-four books were written down, but we know that's not true - many other books were also written down. Rather, I think the Talmud meant that prophecies that were necessary for eternity were preserved and those that weren't - weren't as well cared for, and were thus lost.3
1 Although I know someone who hypothesized in his MA thesis that most of that particular book was copied as-is and inserted in the Book of Kings. One of the appendices of the thesis featured the theoretical recreation of the book. The notion is fascinating, although I'm sure there are those out there who would disagree.
2 It's no less a prophetic work if it was written by someone other than the prophet who prophesied the words in the book. Consider Baruch writing down Yirmiyahu's prophecies and the list of people said to have written down or completed the books of past prophets (Bava Batra 14b-15a).
3 Shmuel Shrirah in his book Mevo Lekitvei Hakodesh, pg. 50, points out that if a canon of books had not been set, then most likely none of the books would have been preserved, considering everything that happened to Am Yisrael during the diaspora. Since a canon had been set, we Jews took great pains to preserve it. We should therefore be thankful for what we do have.