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?