That should be "Read my Bible" as there are many copies of the book ‚not just a single copy which is implied by your use of the ‚which would be correctly used in the sentence "The Empire state building us really tall!" or "Could you direct me to the closest bathroom?"
Reading the bible could be on an IPad or on the internet. (Hey look! Another proper example of using "the" in a sentence!)
Whilst reading my Bible ‚implies ownership of a specific copy of an item.