1 Corinthians 5:1-5

Paul deals with a specific case within the Corinthian church. This case is that "a man has his father's wife." (v1, ESV). You may think that it's his mother even though it's a weird way of saying it, but most are led to believe otherwise. It's actually more likely that his father's wife is his mother in law. In addition, his father probably divorced her or he passed away. Regardless, this is not right and it's even worse that those at the Corinthian church haven't done anything about it. Paul says "Let him who has done this be removed from among you." (v3, ESV).

you are to deliver this man to Satan for the destruction of the flesh, so that his spirit may be saved in the day of the Lord.

(v5, ESV)

This is an interesting verse. It implies excommunication, but some of the wording is a bit weird. You may even argue and question how excommunication would help them. First of all, by doing nothing about it, the Corinthians are basically condoning his behaviour. This is wrong and they need to do something about it. If this were to continue even though he is in the church, he will not be saved living in sin in such a way. The hope is that through the experience of being outside the warm embrace of the church community, this man will experience some sort of hardship or realization. Through this realization, hopefully this man will turn away from his wrongdoing and turn back to God.

I assume that this is only for extreme circumstances. For other situations we can look to Matthew 18:15-18 for if a brother sins against you. There are also a few instructions on rebuking in 1 Timothy 5. Either way, one thing you cannot do is do nothing about it.