Mary Fairchild belongs to the Calvary Chapel denomination, and she writes:
There are no specific scriptural instructions on how to dispose of an
old Bible. While God's Word is holy and to be honored (Psalm 138:2),
there's nothing sacred or hallowed in the physical materials: the
paper, parchment, leather, and ink.
She suggests that, instead of discarding or disposing it, a person should consider donating it to a ministry or someone who wants one.
Christian, Protestant, conservative, evangelical, fundamental, and non-denominational people on gotquestions.org (I think they're Fundamentalists.) write:
There is no "biblical" way to dispose of an old/damaged/ruined Bible,
as the Bible itself does not address this question. It is, therefore,
a matter of personal conviction. It is important to understand that it
is not the paper, binding, and ink that are "sacred" or "holy." It is
the Word of God that is holy, not the material it is printed on. We
are not to worship or idolize the Bible. The purpose of the Bible is
to teach us about Jesus the Savior and the salvation He provides and
to point us to worship Him with all our heart, soul, mind, and
strength. The paper and ink of a Bible are merely the means through
which God communicates His Word to us.
Like Mary Fairchild, they advise donating to the donator's descendants or others who want it.
Catholics agree! I will let them speak for themselves.
There is no specifically mandated means of disposing of old Bibles.
Some Catholics follow a custom of disposing of religious articles that
have been blessed either by burying or burning them, but even that is
not mandated by law. If the Bible has been blessed you might choose to
follow that custom. If not, dispose of it as you would any other book.
If it’s still in fair condition, you might put it on a book donation
table to benefit someone else.
Jon M. Sweeney does not really mention the LDS Church, but he does have a rather strong opinion of what to do about any type of holy book -- be it Christian, Jewish, Muslim, whatever -- that the asker may find helpful, for the reason that the asker belongs to the LDS denomination:
You don't burn them. You never, ever burn them.
You bury them.