Jesus died for your sins
"Jesus died for your sins" (or "Jesus paid for your sins") is a common appeal to emotion used by Christians when attempting to offer a reason to why one should accept Jesus Christ as their saviour. It is a form of Christological argument. In Christian theology, the concept is known as Atonement.
This statement is based upon the Christian doctrine that Jesus was sent to Earth by God to take away the sins of the world, and was crucified, died, resurrected three days later and rose to heaven to be with God, his father (who is also himself).
"He was delivered over to death for our sins and was raised to life for our justification."
- "The reason Jesus had to die for our sins was so that we could be forgiven and go to be with the Lord. "
"The central Christian belief is that Christ's death has somehow put us right with God and given us a fresh start. Theories as to how it did this are another matter."
- "It's just as dumb as the idea of a God coming to life and sacrificing himself to himself to serve as loophole for a rule he created. "
God created people knowing they would sin
The way the story of Adam and Eve is written, God apparently created people knowing that they were likely to sin, and then engineered the circumstances in which they would commit sin. Is it reasonable to blame people for acting in the way that God created them?
"That [God] took revenge on his pots and creations, however, because they turned out badly—that was a sin against GOOD TASTE."
The sacrifice is arbitrary
Why a sacrifice at all? Surely if an omnipotent being did not want to eternally punish people, he would simply not eternally punish them. He makes the rules. Jesus claimed that he could revise the law as he felt necessary, for example with dietary laws Matthew 15:11 . How exactly does God sacrificing himself to himself change the situation?
Substitutionary atonement is immoral
Substitutionary atonement is the notion that a person can suffer in place of another to remove guilt. It is generally accepted concept in Christian theology.  Any reasonable sense of justice dictates that guilt and punishment cannot be simply transferred from one person to another.
Appeal to emotion
A common appeal to emotion is that all human suffering pales in comparison to what Jesus went through for us. Yet was any of his suffering related to what he wanted to accomplish? If the suffering was in some way efficacious for his ends, why didn't God make Jesus suffer a little more, thereby increasing the quality of his sacrifice? We can only conclude by the average amount of suffering Jesus underwent that it was all auxiliary to his main goal, which was simply to die; and then the question is why God didn't offer Jesus as a burnt offering (like Jephthah did with his daughter) or kill him in a way that was a little less painful.
Not much of a sacrifice
For an eternal being, dying for three days is not much of a sacrifice. Jesus was crucified by people who didn't agree with his contemporary blasphemy, knowing that he was a god, was taken off the cross just a few hours later, supposedly died, rose from the dead and is now in heaven. Plenty of people have suffered worse tortures throughout history, and have not gotten to become God in the bargain.
"I have heard some people complain that if Jesus was God as well as man, then His sufferings and death lose all value in their eyes, 'because it must have been so easy for Him' [...] The perfect submission, the perfect suffering, the perfect death were not only easier to Jesus because He was God, but were possible only because He was God. But surely that is a very odd reason for not accepting them?"
Plenty of other mythological figures have also suffered much worse than Jesus supposedly did. A prime example of this is in ancient Greek myth, with the Titan Prometheus. Prometheus protected man and guided his journey toward enlightenment. But when he stole the gifts of fire and knowledge from the Olympian gods and gave them to mortal man, Zeus considered this a betrayal. As punishment, Prometheus was chained to a rock to have his liver picked out by a giant bird every day, then regrew every night just to get picked out again the next day. As Prometheus was immortal, he suffered in this way for eons until finally being set free by the hero Hercules, much longer than Jesus' measly three days. Perhaps if Jesus had truly died and was suffering in hell right now (as Judas Iscariot probably would have to for Jesus's plan to work) and for the rest of eternity, that would be a real sacrifice for us.
Jesus was killed knowing it was temporary
According to Christian accounts, Jesus knew beforehand that he was going to rise again (see Mark 8:31 or John 10:17-18 ), and apparently he even told other people about it (see Matthew 27:63 ). If he didn't die (or just pretended to for a few days), how is it a sacrifice?
Jesus cannot die for our sins since Jesus is divine and cannot die.
If Jesus died in a meaningful way, it raises several theological problems. Did the trinity only have two members for a time?
Timing of the arrival of Jesus is arbitrary
Why wait so long? If Jesus truly is the only path to salvation, then people lived and died for thousands of years with no chance to escape hell. (Note that in the Old Testament, people simply died and entered the grave. There was no mention of any hell. It is only with "gentle Jesus, meek and mild", that we have eternal punishment after death for finite crimes.)
God is demanding "protection" from God
An analogy for this situation - By sacrificing Jesus to die for your sins, God has created a situation akin to pushing everyone into a body of water and then holding up a life vest in front of their eyes. "You can all save yourselves if you will just accept my gift. Otherwise you will drown." He has created the situation of peril, and then demands your love in order to be saved from eternal torture.
Jesus negates personal responsibility for actions
Christopher Hitchens argues for a variant of the argument from justice that attempts to undermine substitutionary atonement. It loosely is: "If you are in trouble, I can pay your debt. I can even serve your sentence in jail. If I feel especially kind I may even walk to the gallows in your place. But none of this will absolve you of personal responsibility for committing the wrong or immoral act." In every case, this is what we see with Christianity. A rapist can have a deathbed conversion, be washed of his sin, and enter heaven with eternal reward. While a person who has led a kind life, but does not believe in Christ, will be tortured eternally. This is not justice. This is people casting their worries and guilt upon an animal, and running it out of the village to die in the desert or wilderness, as in old times. This situation of Jesus dying for our sins is so indicative of it's time: blood sacrifices to absolve wrong. How does something bleeding right your wrong?