Most Protestant Christians don't feel the need to use a special ritual to dispose of worn-out Bibles. The books themselves should be treated respectfully (e.g., not used as doorstops or the like) but when they're unusable they can be thrown in the trash, burned, or whatever. However, disposing of worn-out scripture is a very different matter for Muslims and there is some specific guidance on how to dispose of used Qur'ans.
Disposing of Unusable Copies of the Qur'an
As far as old and unusable copies of the Qur'an are concerned, it is not permitted to burn them unless there is no other way to dispose of them. The great Hanafi Imam, Imam Ibn Abidin (may Allah have mercy on him) states:
"If a copy of the Mushaf (qur'an) becomes old and it is difficult to read from it, it should not be burnt in fire. This is what Imam Muhammad (m: student of Imam Abu Hanifa) pointed out and this is what we take. It will not be disliked to bury it. It should be wrapped in a pure cloth, and a Lahd grave (m: grave that has a incision in the side wall, customary in hot climate countries where the earth is solid) should be dug, because if a Shiq grave (m: grave with a straight opening, common in cold climate countries due to the earth being soft) is dug and the copy ofthe Qur'an is buried, it will entail the soil falling on top of the Qur'an which is a form of disrespect, unless a slab is placed as a roof. .. " (Radd al-Muhtar, 5/271)
In light of the above, there are two methods of disposing of an unusable copy of the Qur'an:
(1) Wrapping it in a pure piece of cloth and burying it respectfully in a place where people (normally) do not walk about In cold climate countries (such as the UK), one may dig a Shiq grave, but a slab should be placed first and over it the soil.
(2) Fastening the Qur'an with a heavy object like a stone and then placing it respectfully in flowing water.
However, burning seems to be acceptable as a last resort. Here's some more info:
The burning of the Quran has angered Afghans in the past, sparking deadly protests in 2010 and 2011. But if discarding a Quran is necessary, there are respectful and acceptable ways to do so, scholars say.
It is important to give a Quran a proper burial, says Omid Safi, professor of Islamic studies at the University of North Carolina and author of "Memories of Muhammad." Any text containing the name of God is sacred in Islam. God is revealed through scripture, and anything associated with writing it has a religious significance, Mr. Safi says.
One could literally bury the Quran, ideally in a place with little foot traffic. Another option is to put the book in a flowing body of water, either letting it sink or be carried away. Regardless of the method, treating the book's destruction with respect is paramount. Safi likens it to a poor man's funeral, where the book might be wrapped in a shroud before being placed in the ground and mourned.
Burning the Quran, however, is also an accepted practice.
"People often ask, 'if it's OK for Muslims to burn the Quran, then why isn't it OK for the US military to do it?'" Safi says. "That's where the question of symbolism is important."
Some say erasing the names of God and his messengers prior to burning the Quran makes it acceptable, but Safi says it's even simpler than that. It comes down to context: Burning the text in a dumpster with trash on a US military base feels less respectful than treating the disposal with reverence in a burial or sacred burning.