It is not God's desire that anyone should have to suffer, but suffering is a consequence of this fallen world. In the New Heaven and New Earth, there won’t be suffering. Christ suffered on the cross, and it's fair to say that Christ's suffering was a consequence of this fallen world. “For our sake he made him to be sin who knew no sin, so that in him we might become the righteousness of God.” (2 Corinthians 5:21) In this verse we see that Christ's suffering was because of our fallen state, but that it was God's will so that we might escape this fallen state.
Share in suffering as a good soldier of Christ Jesus. ... Remember Jesus Christ, risen from the dead, the offspring of David, as preached in my gospel, for which I am suffering, bound with chains as a criminal. But the word of God is not bound! Therefore I endure everything for the sake of the elect, that they also may obtain the salvation that is in Christ Jesus with eternal glory. (2 Timothy 2:3,8-10)
Paul is saying that he continues his missionary work despite his suffering in prison. The suffering itself is likened to Christ's crucifixion which He endured and came out victorious in rising from the dead. This victory was assured; the word of God prophesied it. Likewise, our victory is assured, and the word of God will reach the “elect” despite our suffering as we go. In a sense it is God’s desire that we suffer for the sake of the lost, as Christ did. Paul says in the next chapter:
But understand this, that in the last days there will come times of difficulty. For people will be lovers of self, lovers of money, proud, arrogant, abusive, disobedient to their parents, ungrateful, unholy … Indeed, all who desire to live a godly life in Christ Jesus will be persecuted, while evil people and impostors will go on from bad to worse, deceiving and being deceived. (2 Timothy 3:1-2, 12-13)
The reason the righteous suffer as they try to share the gospel is because there are unrighteous people who will persecute them, trying to prevent the spread of the gospel. Given that a Christian should always be spreading the gospel in one form or another as they “desire to live a godly life in Christ Jesus” (their godly life itself being a witness to the gospel), it follows that there will always be opposition to them, and so suffering.
This is one of the chief reasons for suffering, but not the only one. In 2 Corinthians 1:4 Paul tells us that when we suffer we are then able to comfort those who suffer. Our suffering unites us in this way with one another, strengthening the dependencies we have upon one another. A few verses later, Paul speaks of his trust in God to deliver him from his sufferings. So suffering also strengthens our dependency upon God. Suffering does an amazing job of promoting solidarity in the Body of Christ, the Church, which aids the accomplishment of God's world mission to spread the gospel to all people. The enemies of Christ afflict His Church to destroy it, but simply ends up making it stronger. If the Church underwent great persecution today it would strengthen the Church because false Christians would quickly flee, and all that would remain would be those who are genuine in faith.
There is, however, a sort of “martyr's complex” that is not right. Occasionally you will find Christians who, though they want to share the gospel, they do it in an unloving manner, or their message has some kind of error... but any attempt to correct them is simply seen as persecution, and they foolishly reject attempts to correct them. This is a matter of Christian growth for that person. It may also be possible that a Christian is failing to recognise correction for some outright sin. In 1 Peter 4, Peter confirms what we've been saying; that suffering because of our righteousness is to be expected. But then he says:
But let none of you suffer as a murderer or a thief or an evildoer or as a meddler. Yet if anyone suffers as a Christian, let him not be ashamed, but let him glorify God in that name. For it is time for judgment to begin at the household of God; and if it begins with us, what will be the outcome for those who do not obey the gospel of God? And “If the righteous is scarcely saved, what will become of the ungodly and the sinner?” Therefore let those who suffer according to God's will entrust their souls to a faithful Creator while doing good. (1 Peter 4:15-19)
Christians don't get away with sin; we suffer the consequences in this life just like anyone else. This shows the world that, (to paraphrase Peter), if God is not lenient with those who love Him, how much less with those who hate Him?
So perhaps we can illustrate it like this... imagine the principle of a school whose son attends the school he works in. Each day he watches over his son from the window to his office, and each day the child is bullied by the other students. But rather than intervene, the father watches on. So the son runs to the father for comfort, and says “Father, why is it that you do nothing?” The father says “My son, I know that you are beaten by them. I could have them expelled; but for their sake, show them love and speak to them about Christ. I know that if they will just listen to you, Christ can make a real change to their lives! And they won't be mean people any more. Don't give up on them. I love you...” This is like God the Father. But if that child were to then behave as a bully himself, that father would himself reprimand his son.
Therefore do not be ashamed of the testimony about our Lord, nor of me his prisoner, but share in suffering for the gospel by the power of God (2 Timothy 1:8)