My granny says " Treat it like a family member and bury it"
I would bury it too.
Throwing it in the trash is not something I would do. Since what goes in the garbage? Waste.
Just my opinion.
I read somewhere that the Catholic Church used to burn bibles which contained errors from the press.
OH! Found something that may help. From this very website.
Q: What is the appropriate means to dispose of an old Bible?
A: There is no specifically mandated means of disposing of old Bibles. Some Catholics follow a custom of disposing of religious articles that have been blessed either by burying or burning them, but even that is not mandated by law. If the Bible has been blessed you might choose to follow that custom. If not, dispose of it as you would any other book. If it’s still in fair condition, you might put it on a book donation table to benefit someone else.