I don't know if there is a proper protocol for disposing of old Bibles... interesting question. If you're in a Christian family, you invariably seem to end up with lots of Bibles. People give them to each other, you inherit some, you buy ones that are translations or with commentaries you particularly want. I have somewhere between 6 and 10, I can't remember. I clean out old Bibles that aren't personally meaningful or useful by donating them to local Christian bookstores or thrift stores, or just offering them to friends who are Christian and can use them in some way.
I don't see the point in burning books, but it doesn't particularly offend me that people do that. But it just seems wasteful when there are people who will like books another person no longer wants. I think Americans are far too much a culture that just throws away stuff they don't think is useful anymore. In most other parts of the world, our trash is considered useful for a long time yet. In general, I am for passing books I can't use any longer on to someone who can. And like FS, I find the old, worn-out Bibles from our old family members to be treasured possessions. They meant something to those people, and I feel good being care-takers for those items. I also lug around a bunch of other household goods that I don't really use or like that much, like glassware, that mean a lot because they were our grandparents' and it is the memory that we honor by bringing them out on special occasions.
That said, it isn't about the Bible in particular for me, but rather respect for things and what they mean to other people, as well as the work and resources that go into creating them. I have a collection of books on southern California ethnobotany, for example, from my long-time academic mentor. Even if I wasn't into ethnobotany (I am, but say I wasn't), I would still take care of them and cherish them because they are linked to who he is. One day, it is partly how I will remember him in a visceral way. For whatever reason, things that people loved and put their energy and attention into are linked to how I viscerally remember them- not events, but recalling how I felt in their presence. When I hold items of my grandfather's, I hear his laugh and feel his rough hand holding mine or his bear hug. Rather than remember an event with him, it is like I get a moment of being in his presence, even though he is gone. Because of this, things matter to me in that they are linked to people and I tend to reduce clutter by consuming new items very lightly rather than throwing out old items.