Jewish law has a solution for (almost) everything, and one can learn a lot about Jewish values from it. We learn what to cherish and what to avoid, and how not only to act but how we should think about the world.
One of the earliest things Orthodox Jews learn in childhood is that if we drop a siddur (prayerbook) or a chumash (a Jewish bible) we need to pick it up and give it a kiss. This teaches us at a very early age to not only treat our sifrei kodesh (holy books) with reverence but also with love. As a child gets older and can understand more, we teach him other laws regarding sifrei kodesh including the fact that you must not sit next to a sefer kodesh (you should either hold it in your hands or lap or put it onto a table). If a sefer Torah is dropped during the services in the synagogue, the people present at the time must fast.
When old sifrei kodesh are beyond use, we don’t throw them away or burn them, G-d forbid – we bury them. (This is why when the Nazis burned our Sifrei Torah and sifrei kodesh it hurt so much. We mourn much more than the loss of property – we mourn something much more precious).
The issue of burying old sifrei kodesh is the subject of this article in the Times of Israel. It seems that the problem of what to do with obsolete siddurim and machzorim (regular prayerbooks and special ones for the High Holy Days) has been exacerbated recently since every major denomination of Jewry in America has published a new version of their siddurim in the past few years.
I recently bought a new Koren siddur for myself, after using the Artscroll siddur for the past thirty years or so, since I became Orthodox. As you can imagine there were pages starting to come out of the old siddur. It still sits on our bookshelf. I don’t have the heart to put it into geniza (special storage for books before they are buried). I know I will need to someday, but not yet.