Is there a Church policy or preferred way to dispose of worn out/marked up scriptures?
As far as I can tell, the church has not issued an official policy on disposing of worn out scriptures.
While we love and cherish the scriptures, it is important to remember that it is the “words” of the scriptures that we hold dear versus the actual instrument they are delivered to us upon. There is a difference between “sacred words” vs. “sacred paper, binding, leather, and ink”. In other words, it is the meaning and message that is special, versus the actual paper itself.
If you had a digital version of the scriptures downloaded on your smart phone, would you hesitate to dispose of your old phone and upgrade to another because it contained the scriptures? The phone is simply a device. I understand that some people may feel reluctant in simply throwing away their old scriptures as they may have an emotional attachment to them, this is natural. If your scriptures are worn down to the point where even donating them is not an option, perhaps consider recycling them.
As I type this I am reminded of how the Aaronic Priesthood disposes of old sacrament cups each week. The cups are merely a transportation device, but not a sacred object by themselves.