Do you bury a mus'haf in a special grave plot, when it becomes too badly damaged and or too worn out to repair, out of respect for the words of the Quran? Do you bury any writing that has the name of Allah written in Arabic on it?
You have two options in things that have verses on it or Allah's name :
Burn it completely. This is what Uthman ibn Affan did:
"When Uthman got hold of all of the Qurans that were written in different dialects, and in some cases were altered to sound exactly like the other dialects, he ordered for them to get burnt because they did not use the proper Arabic that was revealed unto our beloved Prophet Muhammad peace be upon him, and in some cases their words were different because of this dialect difference. He then compiled all of the Noble Surahs (Chapters) of the Noble Quran that were already written during the time of Prophet Muhammad in the city of Madina and formed what we call today the one true copy of the "Noble Quran". Refrence
Bury it in a good soil such in a mosque's land. According to Al-Haskafi, he said :
"الْمُصْحَفُ إذَا صَارَ بِحَالٍ لَا يُقْرَأُ فِيهِ : يُدْفَنُ ؛ كَالْمُسْلِمِ"
He says that if a Quran gets weary and can't be read from, it gets buried, like a Muslim. Reference
Old Quran could be sent to Islamic library, Mosque or Madrasa where people will read it and you will get (صدقۂ جاریہ) Perpetual Charity, a deed that always keeps benefiting people, even after the person’s death (building a mosque, water well, Islamic centre/ hifz Quran, planting trees). If it’s not in good condition then below is the last option to dispose the Quran in a way so that its honour and respect is retained.
Islamic teachings about the disposal of Quran largely falls into below these main options, which are all ways of returning the material naturally to the earth: burying, placing it into flowing water, or burning.
BURYING : With this method of disposal, the Quran is to be wrapped in cloth to protect it from the soil, and buried in a deep hole. This should be done in a place where people would not normally walk, often on the grounds of a mosque or even a graveyard. According to most scholars, this is the preferred method.
PLACING IN FLOWING WATER : It is also accepted to place the Quran in flowing water so that the ink is removed from the page.
Burning : Most Islamic scholar agree that burning old copies of the Quran, in a respectful manner in a clean place, is acceptable is a last resort.
This will wipe away the words, and disintegrate the paper naturally. Some scholars recommend weighing down the book or papers (tying them to a heavy object like a stone) and casting them into a flowing river or sea. One should check into local regulations before following this method.
Allah knows best.