First, the most obvious biblical analogy in The Matrix is the whole
theme of Neo being "The One". Neo represents Jesus Christ, who also
came on the scene of a troubled and doomed human race to save the day.
Second, Neo's name means "new", and reminds me that Jesus Christ ushered in the era of the "New Testament", which is the new covenant God made with His creations to supercede that of the "Old Testament".
Trinity's name, also, is blatantly biblical. The Trinity, in the Bible, is sometimes referred to as "the Godhead", and is made up of God the Father, Jesus the Son, and the Holy Spirit. In the movie, Trinity's love brings Neo back to life. God brought Jesus back to life.
That brings me to my next point, which is that Neo is resurrected from the dead in the movie, just as Jesus was resurrected after a 3-day tomb time.
When it would've seemed more logical to let Morpheus die to save the rest of humanity, Neo risks everything to save him. In the Bible, the Father is compared to a man who leaves 99 of his sheep on the hillside to look for the one sheep that wandered away (Matthew 18:12-14). Similarly, Neo leaves everyone on the ship to save Morpheus, just one man.
There is also the whole element of betrayal in the movie, where Neo is betrayed by one of his shipmates. Jesus was betrayed by one of His disciples, Judas Iscariot.
The true homeland of Morpheus, and the others who know the truth, is Zion. In the Bible, Zion is one of the hills on which Jerusalem stood, and is also used to refer to all Jerusalem, to the Jewish people, or to heaven (Hebrews 12:22).
My last analogy is the one that really got me. The Matrix is the world we live in, the world we know. In the movie, it wasn't the truth. It was like the whole human race was living a lie, thinking that their reality was all there was to life. The Bible makes it clear that the world we live in is not "it". There are wars being waged in the spiritual realm, there is an afterlife we cannot fathom, there is a whole other reality that we routinely deny.
Living in the truth, as in the movie, is not an altogether pleasant and painless experience. It is far easier to live in "The Matrix". But the real freedom can only be found when one deliberately chooses "the red pill", which is truth. The "red pill" in our reality is the Bible, and everything in it.