Crucifixion of Jesus

Overview

Christians believe that Jesus was crucified to His death. Muslims generally say that Jesus didn't die on the cross. Who is correct?

The answer is that both are correct. If you focus on the material body, Jesus was crucified; if you focus on the immaterial soul, Jesus was not killed. You can bang nails through a body to kill it, but you can't do the same to a soul. So rather as when you break a cage the bird flies free to the sky, when Jesus' body was destroyed, His soul ascended free into the Heavens of nearness to God.

Jesus 'Appears' to Die

The passage in question causing Muslims to believe that Jesus didn't die on the cross, and that God crucified the innocent Simon Cyrene in His place (or some other conjecture), can be found at 4:157. Ultimately, the reason this passage causes confusion is that it is read without any reference to other statements in the Qu'ran that make the sense quite plain, that Jesus did die on the cross.

4:157 And [the Jews] saying: Surely we have killed the Messiah, Isa son of Marium, the apostle of Allah; and they did not kill him nor did they crucify him, but it appeared to them so and most surely those who differ therein are only in a doubt about it; they have no knowledge respecting it, but only follow a conjecture, and they killed him not for sure.
Nay! Allah raised him towards Him; and Allah is Mighty, Wise. And there is not one of the followers of the Book but most certainly believes in this before his death, and on the day of resurrection he (Isa) shall be a witness against them.

In the Qu'ran, this passage comes against the Jews not the Christians, although it takes them both into consideration. It's important to look at what Jews and Christians believe in order to understand what the Qu'ran is getting at here.
From the Jews' point of view, Jesus was a false teacher leading the Jews astray from a strict adherence to the very complex Jewish customs. They killed Jesus (variously by crucifixion or another means) and depending on a Jew's understanding of afterlife, if any, Jesus was either, utterly destroyed for eternity; or he was consigned into the burning pit of Hell; or he is sleeping and will awake at the Resurrection to discover the error of his ways. Whichever Jewish view you consider, Jesus was permanently dealt with with no further influence on the affairs of men, and had little place with God.
From the Christian point of view, Jesus was a true teacher and was unjustly killed, yet rose from being dead and continued to influence the hearts, minds and structures of the world in a profound way. A complex belief is also held that this process of dying and rising was necessary for Christ to save his followers from the sins introduced into the human sphere by Adam and Eve.
Fundamentally both the Jewish and Christian views suffer from the same mistake of confusing Jesus' soul with his body, which is like confusing the bird with its cage, or a bicycle with its rider. When you realise that killing the body, so far from inhibiting the soul, actually releases the righteous soul to arise to its greater purpose, then the common Jewish, Christian and Muslim doctrines are shown to be the mistaken products of a materialistic world view that sees merely the body. The Qu'ranic passage addresses all these issues and several others.

To paraphrase the passage question, it is simply saying:
The Jews assert, "We killed Jesus", but they didn't kill him, they didn't even crucify him; it appeared so to their outer eyes which see only the body and imagine no further. Those who dispute over such a matter have no real knowledge of reality but follow their materialistic presumptions: for certainty, they did not kill him: certainly not! on the death of his body, Allah raised up the reality of Jesus (his soul) ever closer to Himself through His power.

What the Qu'ran is saying is exactly the way you or I would reassure someone whose loved one had just died, say in a car accident; you would comfort the person by saying: "They weren't really killed, it was their body that was run down by the car, you are shocked because it seems so much that way, yet they have ascended up into the spiritual world and are alive and well, drawing ever closer to God."
This is purely the intent of the passage in the Qu'ran.

Speaking of the Dead as Alive

In particular, various verses of the Qu'ran insist one should never say of those who die on the path of God that they are dead, but to say they are alive, for instance:
2:154 Do not speak of those who are slain in Allah's way as dead; nay, (they are) alive, but you do not perceive.

The Qu'ran is mere applying this standard of speech in its own verses concerning Jesus.

Other Passages on Jesus' Death

Many muslims view Jesus as being raised alive to heaven (presumeably with a view to his returning). Not only is heaven not a place where physical bodies of this earth are to be taken, but the Qu'ran does actually state that Jesus died:

5:117 I [Isa] did not say to them aught save what Thou didst enjoin me with: That serve Allah, my Lord and your Lord, and I was a witness of them so long as I was among them, but when Thou didst cause me to die, Thou wert the watcher over them, and Thou art witness of all things.

3:55 [The Jews] plotted, and God plotted. God is the supreme Plotter. He said: "Jesus, I am about to take you and lift you up to Me. I shall take you away from the unbelievers and exalt your followers above them till the day of Resurrection. Then to Me you shall all return and I shall judge your disputes.

Now some might God would prevent a prophet from being humiliated and killed, as a prophet is representative of God. The Qu'ran does not endorse this view. It presents prophets as suffering immensely, as Muhammad did himself, and also that prophets get killed unjustly (e.g. 3:181, 4:155). In fact, it is a Prophet's dearest wish to be permitted to be martyred on God's path, and by so doing bequeath to mankind an enduring example of submission and sacrifice for the greater will of God.

Some Consequences

There are various consequences to this misreading of the Qu'ran: