Are Christians supposed to judge other Christians in the church?
Yes, Christians are to judge those within the church when it comes to discipline for gross, immoral sin. While general judgment is forbidden, the believing community must confront a member living in sin and point it out, rather than turn away. Judgment of outsiders, however, is not the church's responsibility.
Les explains that in 1 Corinthians 5, Paul distinguishes between judging outsiders and judging those inside the believing community. Paul says he has nothing to do with judging those outside—only preaching the gospel to them—but he asks, 'Do you not judge them that are within?' Les reminds us that Romans teaches we are not to judge, and for the most part that is true. However, when there is gross, immoral sin within the believing community, believers should not simply turn the other way and say, 'I can't judge.' Instead, they are to take the bull by the horns and point out to that person that they are living in sin. Les also references Galatians 6 for further instruction on practical Christian living.
For what have I to do to judge them also that are without? do not ye judge them that are within?