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.
It would seem, though, that throwing away a Bible would have to be the absolute last resort. Many old Bibles, if rebound, would be "as good as new" and could then continue to be used or given to someone else to use. Even an old/tattered Bible, given to someone who does not have a Bible, can be a tremendous blessing. Some people keep old Bibles as family heirlooms to remind future generations of how committed to God’s Word their ancestors were. There are many such options and all are worth prayerfully considering.
Ultimately, if a Bible is genuinely damaged to the point of being unusable, it can be discarded. In no sense will discarding a Bible invite God’s displeasure. Some prefer to burn a Bible rather than placing it in the garbage. Neither method is right or wrong. There is a story of a person who threw a Bible out with the trash. A garbage collector saw the Bible and removed it from the trash, began to read it, and placed his faith in Jesus Christ as Savior. God can use His powerful Word (Isaiah 55:11; Hebrews 4:12) to testify of Him, even from the pages of a Bible that has been discarded.