If being a non-Christian is a sin, and if Jesus died for all of our sins, why should non-Christians go to hell?
A comment you made asks if Jesus died only for Christians, and not for all of humanity. The answer is in John 3:16-17:
For God so loved the world that he gave his one and only Son, that whoever believes in him shall not perish but have eternal life. For God did not send his Son into the world to condemn the world, but to save the world through him.
God loves humanity so much He gave us the most precious gift imaginable – His Son. Jesus laid down his life for humanity but not everyone will believe in his name. The undeserved gift of forgiveness and eternal life is conditional, according to John 3:18:
Whoever believes in him is not condemned, but whoever does not believe stands condemned already because he has not believed in the name of God’s one and only Son.
Jesus said this in John 14:6-9:
I am the way and the truth and the life. No-one comes to the Father [God] except through me. If you really knew me, you would know my Father as well... Anyone who has seen me has seen the Father.
The truth of these statements is extended to any person, regardless of race, colour, ethnic origin or religion. The free gift of eternal life is open to ALL who come to saving faith in Christ Jesus.
On the night before Jesus was crucified, he prayed to his Father in heaven. Part of that prayer is recorded in John 17:3-5:
Now this is eternal life: that they may know you, the only true God, and Jesus Christ, whom you have sent. I have brought you glory on earth by completing the work you gave me to do. And now, Father, glorify me in your presence with the glory I had with you before the world began.
Jesus, the Son of God, has done all that is necessary for people to be forgiven for their sins. The free gift of eternal life is extended to everyone. That is one reason why Christianity is unique. Every other religion says you have to work at pleasing God by doing good works (such as praying five times a day and following the other rules in the Qu’ran) in the hope that you might receive mercy. The Bible says God freely forgives all who repent and place their faith in Christ Jesus, His Son. That free gift is open to every person. It is only those who reject Christ Jesus that face judgment.
Your main question asks ‘why should non-Christians go to hell’. In his answer to you, Matthew said:
Since everyone is a sinner, everyone, Christian and otherwise, stands justly condemned and is in need of Salvation. Scripture is also quite clear that the only Way to Salvation is Christ. Therefore, only Christians have any hope of avoiding the consequences of Sin. It's true that Christ died for the ungodly, but only those that accept Christ can be Saved.
Neither religion or good works can save us. Only placing our faith in Christ Jesus can save us. It really is that straightforward.
It is not being a non-Christian that is a hell-deserving sin. It is not being unable to obey all the commands of God that is a hell-deserving sin. It is not any one of a list of thousands of sins you can think of that means we are hell-deserving. Hell is for sinners, full-stop, and all of us are sinners. Me, you, our next-door neighbours, the religious leaders we hope are guiding us spiritually... everybody from Adam is, by nature, a sinner who is spiritually dead and headed for hell. Our sin alienates us from God our Creator. He is holy, righteous, and just, and cannot look upon sin. The Bible explains that in hundreds of texts, both in the Old and the New Testaments. Yet, despite that, God so loved the world that he decreed this from before Creation began:
"And as Moses lifted up the serpent in the wilderness, even so must the Son of man [Jesus Christ] be lifted up; that whoever believeth in him should not perish but have everlasting life, for God did not send his Son into the world to condemn the world, but that the world through him might be saved. He that believeth on him is not condemned, but he that believeth not is condemned already, because he hath not believed in the name of the only begotten Son of God." John 3:14-18 A.V.
There is the answer to your question in that last quoted sentence. But here's the rub: different religions (and non-religions) portray a false Christ who is not the Jesus Christ of the Bible. The Qur'an, for example, only has about 25 verses in it that mention Jesus, the son of Mary, most of them being a negative portrayal designed to contradict what the Bible stated about Jesus. The Bible was written many hundreds of years before the Qur'an. The accounts about Jesus Christ were written by men who walked and talked with him, who saw him crucified, then resurrected, and 40 days later, ascending bodily back up into Heaven, where he had come from. The Qur'an denies all of that. So, people have to decide which Jesus they believe in. To believe in a false Jesus is to believe in a false gospel, which keeps sinners on the pathway to hell. There are also many who call themselves Christians who are in that deceived company. The Bible is clear in its warnings about that too.
So the really important question is not, "Why would non-Christians go to hell", but "Why would anybody go to hell?" and once you've got the Bible's answer to that, the matter should open up to you in a saving way.
First off, not being Christian is not in and of itself sinful.
What's sinful is failing to keep God's commands, which include recognizing God as the sovereign of Creation and doing evil of any kind.
No human ever Created is not sinful. No one (except Christ) has ever kept God's Law fully. This is the important part, and Scripture is quite clear on this point (e.g. 1 John 1:8,10, Romans 1-3). Since everyone is a sinner, everyone, Christian and otherwise, stands justly condemned and is in need of Salvation. (Note, however, that there is disagreement on what form that punishment will take. See here, for example.)
Now, Scripture is also quite clear that the only Way to Salvation is Christ. Therefore, only Christians have any hope of avoiding the consequences of Sin. It's true that Christ died for the ungodly, but only those that accept Christ can be Saved.
Which sins didn't Jesus die for? Did Jesus only die for Christians, not for humankind?
God's plan of salvation through Christ is a bit like a bowl of "get out of hell free" tickets. Anyone can take a ticket; in that respect, Jesus died for everyone. However, not everyone does take a ticket; those that do are "Christians", and in that sense, Christ only died for Christians.
Alternatively, imagine a bowl of tickets for free ice cream in a park with exactly 1,000 people. In theory, to cover those tickets, God would need 1,000 servings of ice cream. However, God, being omniscient, knows exactly how many people will actually take a ticket and brings exactly the right amount of ice cream. This is the sense in which people will say that Christ died only for certain people, or certain sins. However, the reason some people don't get ice cream (that is, are not saved) isn't because God didn't bring enough, but because they didn't take a ticket (didn't accept Christ).
About the inclusivist view
First of all, Christians have different views on the destiny of non-Christians, especially of those who through no fault of their own never heard or understood correctly the gospel of Jesus Christ (see my long answer to another question in which I included a chart of 5 views on the Destiny of the Unevangelized).
Then, there is the matter of properly understanding "hell". Among other connotations, for Christians, "hell" is primarily a state of willfully rejecting the offer of life given through Christ which implies rejecting God's gift of felicitous fellowship with God through being united with Christ, which can be caused by several reasons including wanting to conduct life on earth according to his/her own standard and relating to God according to his/her own method, despite already understanding the better way given by Christianity.
A Christian's way of relating to God (which leads to heaven) is one of:
- pure dependence as creature, which means living according to God's standard of human nature
- pure loving God as gratitude of being given not only biological life but spiritual life
- pure loving our neighbors including our enemies following Christ who died for His own enemies, and
- willingness for "being yoked with Christ" in a lifelong process of transformation by the Holy Spirit so gradually sinful desires become extinguished, a painful mental process that in Christian lingo is given the name "carrying one's cross".
In the inclusivist view, going to hell is a matter of active rejection with full understanding. So it is possible for a Muslim who rejects a distorted presentation of the gospel to not go to hell if he/she is rejecting the wrong idea of Christ and the life He offers. Christians who believe in the inclusivist view believe that EVERYONE (non-Christian or not) will have the opportunity to respond to this offer of life, either
through the preaching of the gospel in a church or through a missionary activity in the field, or
given imperfectly through another religion, with the Holy Spirit acting as a missionary
- guiding the individual's heart to increase the desire toward the unknown Christ, manifested in his/her dissatisfaction with their own religion, or a wondering about God's mercy and love not taught in their own religion
- giving the grace to do to God and to others whatever their religion is teaching, as long as that teaching is consistent with Christian morality
through a post-mortem evangelization where Christ himself will present the gospel to the individual.
So for the Christian inclusivists, it's not an automatic scenario that all non-Christians go to "hell".
Answering your question from the inclusivist view
Having said that not all non-Christians go to hell, let me address your other 2 points in your question & comment:
If being a non-Christian is a sin, and if Jesus died for all of our sins, why should non-Christians go to hell? So which sins didn't Jesus die for? Or did Jesus only die for Christians? Not for humankind?
In Christian view (whether it is inclusivist or not), sin is an act of rebellion against God. Being a non-Christian in itself is not a sin, especially for those who never heard, let alone, understand the gospel. Of course, once someone understands the gospel enough but STILL rejects it, then at that point it becomes a sin. Christians (whether inclusivist or not) believe that within all human beings there is STILL the light of the conscience, which is God's way whispering to a person not to do evil, especially obvious ones such as killing an innocent person, or intentionally inflicting permanent harm upon another.
About whether Jesus died for non-Christians, this is an internal debate within Christianity. Some say that Jesus died only for the elect (i.e. those who accept Jesus as their savior). Some say that Jesus died for everyone, including those who reject Him. See John Piper's answer to the question In What Sense Did Christ Die for the Non-Elect who argues that Jesus dies for all humanity but in a more special sense for the elect.
Dying to give us life vs. paying for our sins
Please note that the right way to think about Jesus's death is that He dies for more than paying for our sins. Jesus dies a heroic death giving his life for me so I can have life, not simply paying for my sins. It's an exchange of life for life. Hence the logic of owing our lives to Christ after receiving the payment, as a life in gratitude.
In Christianity, Jesus is said to die like we die for our friends, except that He dies while we are still enemies to God, so Jesus shows greater love to us than any of us show love for our friends (John 15:13, Romans 5:10). Jesus wants all humans to be his friends. When we die (as heroes) to save our family / friends, we die for their whole persons despite of their sins. This is how we primarily should see Jesus's death on the cross, with the added benefit that Jesus also died for our sins.
Jesus's death should primarily be seen as enabling us to become sons of God, but this gift must be received, with the Holy Spirit's help to lower our pride to want to receive it. Furthermore, in Christianity, the point of being a Christian is to be free to do what is right, not simply to accept Jesus's sacrifice as payment for our sins; once we receive God's life it should become easier to do what is right.
Let's try a more, simple answer. All man's sins were brought upon Christ when he died for us all. Christian, Jew, Pagan they all had their sins washed in his blood. The question you seem to struggle with is what about a non-Christian? Do they go to hell?
Well man that short answer will be yes but... We all get the choice here. You do not have to be Baptized (though it is a good thing). You do not have to publicly profess your beliefs in Christ. You can believe quietly if you want. But you really need to understand that through Christ you come to The Lord. Many "religions" are really Cultural, not Spiritual. I do not think Buddhists think that he is/was "God" but that he had a way to live a good life. Cool. But you can still accept Jesus as your savior. Right?
The Quran mentions Jesus a bunch. As that same Book lists Mohammad as just a profit there too you can accept that Jesus is the Savior while living the rules of the Quran.
It is really pretty simple. Let me close with a link. The CS Lewis organization does a nice job of explaining it.