The oldest physical copy of any of the Bible are two scrolls which are dated to about 650 BCE. They were found in a burial chamber in the Ketef Hinnom archeological site in the Jerusalem area. They contain the Priestly Blessing found in Numbers 6:22-27.
The next oldest physical texts of the Bible are the Dead Sea Scrolls the oldest of which date to 200 BCE. The over 900 documents include fragments of every book in the Hebrew Bible except Esther. The texts include a scroll that contains the complete book of Isaiah (apart from some portions that are too damaged to read).
The oldest known religious writings (as opposed to say, cave paintings, which almost certainly had some religious significance) are The Pyramid Texts, which date back to the Old Kingdom (3rd millennium BCE).
They were probably written in 2400-2300 BCE, and detail the ancient Egyptian perception of dying, the afterlife, and resurrection.
Not the oldest, but a rather intriguing text dating back to 1000-750 BCE is the I Ching or Yi Jing.