First I should probably clarify a few things. I am not trying to promote a fundamentalist view of scripture here. That is why I deliberately differentiate between a bible and the Bible. If I am talking about a specific bible, as in the book on my shelf or the one in the drawer of your hotel room, I use "bible" because we are talking about a book. However, if I am talking about the Bible, as in the Word of God, I use "Bible," because we are talking about God's Word collected in scripture.
Second, I am not saying bibles need to be disposed of in a special ceremony like old American flags or something like that. What to do with our old bibles becomes a question when we realize we are talking about getting rid of a book that happens to contain God's Word, and when we realize that we are talking about a book that people have spent time reading and learning from everyday for a long period of time. People develop emotional attachments to their bibles. (This is partially due to the fact that it is The Bible-it is God's revealed Word.) So getting rid of a bible can bring feelings of guilt or remorse.
So what do we do with our old bibles/Bibles? I think this really depends on the condition of the specimen. If the bible is still in relatively good condition I would recommend donating it to your church or a new and used store or some other charity. This will allow for the chance that this bible, which also happens to be the Bible, to impact the life of someone else. There are many organizations that would love to receive your used bibles and share them with others.
If your bible, however, is very used (i.e. a missing cover, or highlighting on every page), I would suggest recycling it, or throwing it away. These circumstances force us to realize that this is only a bible-a book. There is not some intrinsic value to the printed words themselves. It is the message contained in the book that has the power to change hearts and not the book itself. Indeed, if it were the specific words printed on those onion pages that were holy, we would not be allowed to translate the Bible into different languages because we would run the risk of loosing some of the meaning contained in the original Greek, Hebrew, and Aramaic. No, it is not the words themselves that are holy and sacred but the message of forgiveness of sins. It is this promise that has its origin in God's grace, not the ink from the printer or the vocabulary choice of the translator.
So, if you have an old bible do with it what ever you see fit. Just remember that while your bible contains God's Word, the marks on its pages are not Holy in and of themselves.

