If someone is forced to take the punishment for sin (Jesus), doesn't that mean that God never forgives?

You didn't specify the denomination whose take you want, so I'm answering from a Catholic perspective.

God forgives because Jesus suffered to merit our freedom from suffering (Jesus was not 'punished'—Jesus did no sin, so His suffering for our sin is meant in some other way than 'punishment;' namely, the greatest form of penance, or suffering, to the procuring of graces, made so effective in the perfect love with which it was offered, and His infinite dignity as the incarnate Son). The prefiguring sacrificial lamb was sacrificially offered to propitiate; so with Christ: the Lamb of God was not 'punished,' (as if he did the sins of the people, but rather their sins were 'imposed upon' the lamb symbolically by the imposition of hands) but substituted our suffering in Hell, with His perfect fulfillment of God's justice, and thus the reward of ransoming people who don't deserve it save His pure sovereign mercy and choice.

Jesus freely offered to satisfy the demands of justice by radically uniting with us (in a much more real way than we think when we usually use the word 'union' between God and man) so that His suffering could be offered in our stead—instead of our eternity in Hell, He offers the equivalent satisfaction: His Passion (it deriving its equivalency—in reality overflowing well beyond the need—in the fact that He is the Creator of infinite dignity, not only a man).

The alternative is unjust forgiveness (when forgiveness becomes a synonym with 'not having justice at all'). In this case, no consequence, not exacting of any sort is made on sin. God's justice is mocked, and so is not justice (where such mockery and insult is itself unjust). The alternative would be us suffering for our sins, which would result in Hell, because each sin which is deliberately committed against God Almighty is equivalent to willing Hell, and separation from Him, because "[God] will not enter into a malicious soul, nor dwell in a body subject to sin" (Wisdom 2:4). Which is Hell, or to be dead even while you are alive (1 Timothy 5:6). All reconciliation is from God, then. "It is not of him that willeth, nor of him that runneth, but of God that sheweth mercy" (Romans 9:16)—we depend utterly on God not to go to hell, because "everyone has sinned and fallen short of the glory of God" (3:23).

In no version of Christianity was Jesus 'forced' to become man and to suffer.

Philippians 2:1-11 (DRB)

1 If there be therefore any consolation in Christ, if any comfort of charity, if any society of the spirit, if any bowels of commiseration: 2 Fulfil ye my joy, that you may be of one mind, having the same charity, being of one accord, agreeing in sentiment. 3 Let nothing be done through contention, neither by vain glory: but in humility, let each esteem others better than themselves: 4 Each one not considering the things that are his own, but those that are other men's.

5 For let this mind be in you, which was also in Christ Jesus: 6 Who being in the form of God, thought it not robbery to be equal with God: 7 But emptied himself, taking the form of a servant, being made in the likeness of men, and in habit found as a man. 8 He humbled himself, becoming obedient unto death, even to the death of the cross. 9 For which cause God also hath exalted him, and hath given him a name which is above all names: 10 That in the name of Jesus every knee should bow, of those that are in heaven, on earth, and under the earth: 11 And that every tongue should confess that the Lord Jesus Christ is in the glory of God the Father.

John 10:17-18 (DRB)

Therefore doth the Father love me: because I lay down my life, that I may take it again. 18 No man taketh it away from me: but I lay it down of myself, and I have power to lay it down: and I have power to take it up again. This commandment have I received of my Father.

Right from the time of Abel it was made clear that God must be propitiated with a victim, specifically a life (Genesis 4:3-5). This means forgiveness is not the unjust kind I described above—sweeping sin under the heavenly rug, as it were: no need for sacrifice. But it is the kind whereby He is willing to have the satisfaction of His justice made by a pleasing victim. Of course things of His creation are not what satisfy His wrath against sin, but these foreshadowed His Son, who can and did.

In other words, 'forgiveness' never means 'ignoring sin,' which would be a lie, but rather recognizing that there is a sin-bearer for whose sake to grant someone remission of their sins, Him having bore them (Isaiah 53:12).

As St. Paul writes, God only showed mercy, not as ignoring literally the sins, but on account of Christ:

Romans 3:25-26 (DRB) [Jesus] God hath proposed to be a propitiation, through faith in his blood, to the shewing of his justice, for the remission of former sins,* through the forbearance of God, for the shewing of his justice in this time; **that he himself may be just, and the justifier of him, who is of the faith of Jesus Christ.

St. Paul is saying God is showing Himself not unjust for passing over sins, which would be unjust technically, by presenting to the world His Son, "the Lamb of God," who bears those sins. The sin is borne, taken away (John 1:29), not ignored (the sense in which too many people are using the word 'mercy').

What meaning does mercy really have, then? Well, clearly not ignoring the problem of sin, as with men whose it is not to pardon or impute eternal guilt anyway, but rather it means that those who are justly to be condemned are now, by whatever means (here Christ), not to suffer any longer as they were to, because justice has been otherwise satisfied. This is the sense in which the New Testament uses the word.

What is apparent in the revelation given in scripture is that God is righteous. The righteousness of God is a key feature of Paul's and of Peter's doctrine.

The entire gospel is concerned with the righteousness of God and with the manner in which God, being righteous, has yet fulfilled his purposes in humanity, despite sin and despite transgression, that is to say despite the very existence of that which is contradictory within creation to God himself and despite the contradictory activity of created beings.

The idea that God could ever simply ignore sin and transgression (which is what is being suggested) is completely foreign to all that is revealed in scripture of whom God is and of what he is.

Was Jesus forced to take the punishment for sin? No, he was not. Jesus came willingly to earth to humble himself and take on human form so that he could atone for our sins. Nobody forced him. Jesus himself said that he laid down his life willingly:

“No one takes it from me, but I lay it down of my own accord. I have authority to lay it down and authority to take it up again” (John 10:18).

Even if Jesus is God Himself... that would still mean that He never forgives? Yes, Jesus is God incarnate but no, that does not mean God never forgives. And since Jesus is God, are you suggesting that God forced himself to take the punishment for sin? Not sure I follow the logic there.

God would rather punish Himself than show forgiveness for sin, right? Wrong. Forgiveness for sin is a free and undeserved gift from God to all who place their faith in what Jesus did to pay the price of sin:

“This is love: not that we loved God, but that he loved us and sent his Son as an atoning sacrifice for our sins” (1 John 4:10).

“For it is by grace you have been saved, through faith—and this not from yourselves, it is the gift of God—not by works, so that no one can boast” (Ephesians 2:8–9).

God cannot let sin go unpunished because God is holy, righteous and just. To bear our own sins would be to suffer God’s judgment in the flames of hell. That is why the Father sent the Son (John 7:33), the Son accomplished the Father’s will (John 17:4), and the Son died for sinners (Romans 5:8).

Were God to lay aside His wrath at sin and not give us what sin required, He would cease to be just. But, in His great love and mercy, God provided a way for justice to be satisfied and salvation to be extended:

“God made Him (Christ), who knew no sin, to be sin for us that we might become the righteousness of God in Him” (2 Corinthians 5:21).

God’s perfect justice and perfect mercy met at the Cross. Sin and injustice were punished on the cross, with the Son of God receiving the condemnation for sin. It’s because the penalty of sin was satisfied through Christ’s sacrifice that the Father can extend His mercy to undeserving sinners. God was just in punishing sin, and He can also justify sinners who receive Christ by faith:

This righteousness from God comes through faith in Jesus Christ to all who believe. There is no difference, for all have sinned and fall short of the glory of God, and are justified freely by his grace through the redemption that came by Christ Jesus. God presented him as a sacrifice of atonement, through faith in his blood. He did this to demonstrate his justice, because in his forbearance he had left the sins committed beforehand unpunished – he did it to demonstrate his justice at the present time, so as to be just and the one who justifies those who have faith in Jesus” (Romans 3:21-26).

At the cross, God’s justice was meted out in full (upon Christ), and God’s mercy was extended in full (to all who believe). God’s perfect mercy was exercised through His perfect justice.

Not only is God loving, merciful and forgiving, He is also just. God will not allow sinful deeds to go unpunished:

• He loves righteousness and justice; the earth is full of the steadfast love of the Lord. (Psalms 33:5)
• Righteousness and justice are the foundation of your throne; steadfast love and faithfulness go before you. (Psalms 89:4)
• I said in my heart, God will judge the righteous and the wicked, for there is a time for every matter and for every work. (Ecclesiastes 3:17)
• For the wrath of God is revealed from heaven against all ungodliness and unrighteousness of men, who by their unrighteousness suppress the truth. (Romans 1:18)
• Beloved, never avenge yourselves, but leave it to the wrath of God, for it is written, "Vengeance is mine, I will repay, says the Lord." (Romans 12:19)
• For he is God's servant for your good. But if you do wrong, be afraid, for he does not bear the sword in vain. For he is the servant of God, an avenger who carries out God's wrath on the wrongdoer. (Romans 13:4)
• For the wrath of God is revealed from heaven against all ungodliness and unrighteousness of men, who by their unrighteousness suppress the truth. (Colossians 3:35)

At the same time, God is loving, and as such He does not want us to suffer. In keeping with His loving and just nature, God chose to willing take on human flesh and take the punishment that we all deserve:

• Say to them, As I live, declares the Lord GOD, I have no pleasure in the death of the wicked, but that the wicked turn from his way and live; turn back, turn back from your evil ways, for why will you die, O house of Israel? (Ezekiel 33:11)
• For God so loved the world, that he gave his only Son, that whoever believes in him should not perish but have eternal life. (John 3:16)
• He made the one who did not know sin to be sin for us, so that in him we might become the righteousness of God. (2 Corinthians 5:21)
• And you, who were dead in your trespasses and the uncircumcision of your flesh, God made alive together with him, having forgiven us all our trespasses, by canceling the record of debt that stood against us with its legal demands. This he set aside, nailing it to the cross. (Colossians 2:13-14)
• The Lord is not slow to fulfill his promise as some count slowness, but is patient toward you, not wishing that any should perish, but that all should reach repentance. (2 Peter 3:9)

Because Jesus is God, who is eternal, and lived a sinless life, He is capable of justly paying the price for our sins in our place. Additionally, His sacrifice parallels Adam's original sin; where as Adam's trespass led to condemnation for all men, so Jesus's one act of righteousness leads to justification and life for all men:

Therefore, just as sin came into the world through one man, and death through sin, and so death spread to all men because all sinned— for sin indeed was in the world before the law was given, but sin is not counted where there is no law. Yet death reigned from Adam to Moses, even over those whose sinning was not like the transgression of Adam, who was a type of the one who was to come.

But the free gift is not like the trespass. For if many died through one man’s trespass, much more have the grace of God and the free gift by the grace of that one man Jesus Christ abounded for many. And the free gift is not like the result of that one man’s sin. For the judgment following one trespass brought condemnation, but the free gift following many trespasses brought justification. For if, because of one man’s trespass, death reigned through that one man, much more will those who receive the abundance of grace and the free gift of righteousness reign in life through the one man Jesus Christ.

Therefore, as one trespass led to condemnation for all men, so one act of righteousness leads to justification and life for all men. For as by the one man’s disobedience the many were made sinners, so by the one man’s obedience the many will be made righteous. Now the law came in to increase the trespass, but where sin increased, grace abounded all the more, so that, as sin reigned in death, grace also might reign through righteousness leading to eternal life through Jesus Christ our Lord. (Romans 5:12-21)