As far as I know, getting rid of the Bible is not a big deal - this is because the physical book itself is not an object of veneration, nor is it seen to have spiritual value other than the message it carries.
The Quran, on the other hand, is seen as embodying the message of Allah as it was revealed in its original form to Mohamed, his final prophet. So, when the latest round of violence erupted, BBC World Service interviewed an imam and this is a paraphrase of what he said: the Quran is important and holy but that does not mean it can't be disposed of or that if someone handled it wrongly, you must kill him. The acceptable way of disposing of the book is actually to burn it, but to observe some simple rites, e.g. saying a prayer before burning it, burning it by itself (as opposed to together with junk) and burying the ashes afterwards.
My advice to you if you are faced with that situation is to give it to a Muslim and ask him to handle it for you.