What happened to the Book of Mazarbul?

Question

In The Lord of The Rings: The Fellowship of the Ring, the Company that set out from Rivendell with the Ring took the road through Moria, guided by Gandalf. Along the way, they encountered the Chamber of Mazarbul, where Balin's tomb was. At that time, Gandalf read excerpts from the Book of Mazarbul that was also found in the room. Later, he gives it to Gimli to keep.

"We will take this book, the Book of Mazarbul, and look at it more closely later. You had better keep it, Gimli, and take it back to Dain, if you get a chance. It will interest him, though it will grieve him deeply." -Gandalf to Gimli in The Lord of the Rings, Book Two (The Bridge of Khazad-dum)

Gimli obviously cared very much for Balin and grieved that he had died.

Aragorn picked up Frodo where he lay by the wall and made for the door, pushing Merry and Pippin in front of him. The others followed; but Gimli had to be dragged away by Legolas: in spite of the peril he lingered by Balin's tomb with his head bowed. - The Lord of the Rings, Book Two (The Bridge of Khazad-dum)

So it doesn't seem likely that he would have abandoned the book that recorded Balin's last days. However, no mention of it was made after the scene in Moria.

What happened to the Book of Mazarbul?

Answer 1

It's never discussed after this, in any known writings.

One possibility is that Gimli left it at Parth Galen, when Aragorn says (emphasis mine):

'I will follow the Orcs,' [Aragorn] said at last. 'I would have guided Frodo to Mordor and gone with him to the end; but if I seek him now in the wilderness, I must abandon the captives to torment and death. My heart speaks clearly at last: the fate of the Bearer is in my hands no longer. The Company has played its part. Yet we that remain cannot forsake our companions while we have strength left. Come! We will go now. Leave all that can be spared behind! We will press on by day and dark!'

^{The Two Towers Book III Chapter 1: "The Departure of Boromir"}

However, as you say, this seems unlikely. More plausible is that Gimli eventually took it back to Erebor, at which point it presumably occupied a library shelf for the next few hundred (or thousand) years.

In-universe, however, we can presume that they eventually survived to the present day. Tolkien reproduced some pages from the Book, which were intended to be included in the published Lord of the Rings¹:

And the essay "Of Dwarves and Men", written in 1969ish and published in The Peoples of Middle-earth, is said to have been partly based on the Book (emphasis mine):

This long essay has no title, but on a covering page my father wrote: An extensive commentary and history of the interrelation of the languages in The Silmarillion and The Lord of the Rings, arising from consideration of the Book of Mazarbul, but attempting to clarify and where necessary to correct or explain the references to such matters scattered in The Lord of the Rings, especially in Appendix F and in Faramir's talk in LR II².

^{The History of Middle-earth XII The Peoples of Middle-earth Part 2: "Late Writings" Chapter X: "Of Dwarves and Men"}

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¹ They were not, but were eventually published in Pictures by J.R.R. Tolkien in 1979

² Christopher Tolkien goes on to note that "Faramir's talk" refers to Farmir's abridged history of Gondor and Rohan, in Book IV Chapter 5: "The Window on the West"