For example, Christ doesn't heal a Christian when it's time for them to go to heaven (Psalm 139:16). It's obvious from practical experience that Christ doesn't heal every Christian of everything, because all Christians age, get sick, their bodies die, and they go to be with the Lord. Sin's curse still rests on this material world, in spite of Christ's death. The earth still groans. Christians groan with it (Romans 8:22-23). There were a few extraordinary cases in Scripture where a believer got to go to heaven without dying (Enoch, Elijah), but the rest of us age, wear down, and most of us will die of natural causes.
The Lord doesn't heal when he is punishing sin. King Uzziah sinned grievously by intruding into the sacred role of the priests. Uzziah should have known better! In fact, he did know better, which is why God smote him so sternly. I do not doubt that God forgave Uzziah, but the text says God never took the leprosy away (2 Chron. 26:27). God could have healed Uzziah (He did for Miriam in Numbers 12:10-15), but He did not. Uzziah died a leper.
Any sin can be forgiven, but the punishment of the sin might remain. Sin , especially sin by a leader, always carries serious consequences, and some of those consequences involve physical ailments. Paul said in 1st Corinthians 11:30 that some of the Corinthian congregation were weak and sick, and some had died, because of their selfish mistreatment of others in the flock and the way they dishonored the Lord's Table.
Christ doesn't heal while He is testing faith (Job 1-2). Not every sickness is punishment for sin, and we commit the sin of Job's judgmental friends if we assume this. It's a sin to just automatically assume a sick Christian is in unbelief or did something wrong. God Himself, in Job 1, said Job was a man of excellence, guiltless, so teachings that Job somehow brought his illness upon himself are wrong. Rather, God was vindicating His own praise of Job's faith, and shaming the devil for claiming Job was a religious mercenary, although Job never knew that (and God didn't tell him).
James 5:11 promises God will reward us as we come through His tests in faith. Job's example shows us that illness is one of those tests, along with natural disasters (lightning killed some of Job's herds, a windstorm collapsed a house on his children) and crime (gangs of thieves murdered Job's workers and stole all his flocks). But the Lord eventually stops the test, and rewards the believer.
Christ doesn't heal the payback for foolishness. Not everything bad is from the devil. Some of our problems are self-inflicted. Proverbs 3:18 says wisdom is a tree of life and blessing, so what fruit do you imagine foolishness grows? Elsewhere in Proverbs, Solomon says that drunks get bruises, lazy people get poverty, adulterers get beaten up by jealous husbands, and schemers fall into the pits they themselves have dug. Christians could catch AIDS because of immorality, and they shouldn't count on Christ miraculously healing them of it.
Christ knew not to throw Himself off the temple mount on the notion that God would catch Him, because you don't try to force God's hand by deliberately doing stupid, reckless things (Luke 4:9-12). Christ will let us live with the physical consequences of our own foolishness.
Christ doesn't take away the physical scars of persecution. In Galatians 6:17, Paul reminded his readers he still carried in his body the scars of Christ -- referring to wounds left on him by persecution. Paul suffered physically in his service to Christ. In 2 Corinthians 11:25, Paul said he had been pelted with stones and beaten with rods for the Gospel. He still had those scars. Christ allowed Paul to carry those badges of faithful honor in his body, like the medals of a military hero.
Psalm 103 doesn't contradict these other passages. Instead, we take them all side-by-side, believe all of them, and, with the help of other sound Christian teachers and thinkers, think through the best way to harmonize them. Unlike the saving forgiveness of sin, God makes certain exceptions to His normal, default work of ministering to His people's diseases. In the end, all of God's people will be forever immortal, vital, and whole through the redeeming work of His Son.