This is a good question. And the answer depends upon the extent of the damage. Any serious student of God's Word is bound to have the occasional mishap. Being that I have cycled through quite a number of copies of the Bible in my years, I have been faced with a dilemma: what should be done with copies of the Bible that are worn or damaged beyond repair?
Do you 1) store them away somewhere forever? or 2) throw them away? If you're like me, the thought of tossing God's word out with the garbage is out of the question. And unless that copy of the Bible has real sentimental value (gift from loved one, family bible, etc.) storing somewhere may also be an undesirable option. So there you are with a Bible that you can't really use or get rid of.
Allow me to at this point to make a humble suggestion: Burn it.
This may seem like the last thing you would ever want to do with a Bible, but allow me to explain. When in the military, we were trained to have a high reverence for our country's flag (and rightfully so). And when a flag became worn and tattered from use, flag loving Americans face the same dilemma: storing them all away somewhere is impractical and throwing them away is out of the question, so what do we do? Well, at some point during the year, all of the retired flags would be gathered up and they would hold a brief service where the flags would be ceremoniously burned.
Because of my exposure to this method of disposing a reverenced and sensitive item, I have come to see this as an appropriate means of 'decommissioning' Bibles that are damaged or worn beyond repair.
If you have a fireplace, you may consider doing it during family worship. You could explain to the kids the sacredness of Holy Scripture and reasons for disposing of it in this way. Then you might lead them in a prayer or hymn of thanksgiving for God's Word.
This is what we have done in the past.