What would God do if the man in 1 Corinthians 5 kept living with his stepmother?

God would take the man out of Corinth—an early physical death—because his sin was unto death. If he continued unrepentant in the marital relationship with his stepmother, God would remove him, delivering him to Satan for the destruction of the flesh so his spirit could be saved.

Les explains that all unrighteousness is sin, whether little or great, but there is a sin not unto death. The implication is that some sin is gross enough that God will take that believer out ahead of time. In 1 Corinthians 5, the man is evidently committing a sin unto death—worthy of death if he does not repent and turn around. If he continues living with his stepmother in a marital relationship, God is going to take him out of Corinth. Les ties this to verse 5: such a one is delivered unto Satan for the destruction of the flesh, so that the spirit or soul may be saved in the day of the Lord Jesus. Les also notes the Corinthians were glorying and puffed up, and a little leaven leavens the whole lump. He applies this to the church accommodating social sins rather than causing a ruckus.

1 Corinthians 5:1

It is reported commonly that there is fornication among you, and such fornication as is not so much as named among the Gentiles, that one should have his father’s wife.

Who was the man living with his father's wife in the Corinthian church?

The supplied passage does not provide a personal name; it identifies the offender as a stepson living with his stepmother as husband and wife. The Corinthian congregation knew of the gross immorality yet remained puffed up instead of mourning and removing him from fellowship.