Parts of this are from an article I found here

Judge not, that you be not judged. For with what judgment you judge, you will be judged; and with the measure you use, it will be measured back to you. And why do you look at the speck in your brother’s eye, but do not consider the plank in your own eye? Or how can you say to your brother, ‘Let me remove the speck from your eye’; and look, a plank is in your own eye? Hypocrite! First remove the plank from your own eye, and then you will see clearly to remove the speck from your brother’s eye. (Matthew 7:1–5, NKJV)

Notice that the point of Jesus’ teaching here is that we are not to be self-righteous or hypocritical in our judgment. For example, if I was an alcoholic, then I have no business telling someone how sinful they are for being an alcoholic. That’s hypocritical. However, if I do not have a “plank” in my own eye, then I can see clearly to remove the speck from my brother’s eye. In other words, if I am living a godly life, then I can be in a position where I can properly point out sin in another person’s life. This should be done with gentleness and respect, rather than self-righteous judgmentalism.

There are several problems with someone citing this verse as “proof” that a Christian should never try to point out sin in someone’s life. First, later in the same chapter Jesus said, “Beware of false prophets, who come to you in sheep’s clothing, but inwardly they are ravenous wolves…by their fruits you will know them” (Matthew 7:15, 20, NKJV). How can you possibly know who is a false prophet unless you make judgments about their “fruits” (teaching, actions, etc.)? Obviously, we must make judgments in these cases.

Second, Jesus and the apostles demonstrated on numerous occasions that we need to make judgments of sinful behavior. Jesus showed mercy to the woman caught in adultery, but then He said to her “Go and sin no more” (John 8:11). When Simon the sorcerer thought he could buy God’s gift of the Holy Spirit with money, Peter said, “Your money perish with you, because you thought that the gift of God could be purchased with money! You have neither part nor portion in this matter, for your heart is not right in the sight of God” (Acts 8:20–21, NKJV). Paul publicly rebuked Peter for leading people astray by slipping back into a form of legalism (Galatians 2:11–21). So when professing Christians cite Matthew 7:1 out of context, they reveal their ignorance of the examples set by our Lord and His apostles.

Third, notice the absolute hypocrisy of the person who tells you not to judge. What did they just do when they told you not to judge? They judged you. They made a judgment that what you were doing was wrong, and then they went ahead and hypocritically told you that you shouldn’t ever tell people that they are wrong.

Then there are these verses:
2 Timothy 4:2 & 3
Preach the word; be instant in season, out of season; REPROVE, REBUKE, exhort with all long suffering and doctrine. For the time will come when they will not endure sound doctrine; but after their own lusts shall they heap to themselves teachers, having itching ears;

Galatians 6:1
Brothers and sisters, if someone is caught in a sin, you who live by the Spirit should restore that person gently. But watch yourselves, or you also may be tempted.

1 Timothy 5:20
Those who continue in sin, rebuke in the presence of all, so that the rest also will be fearful of sinning.
(that one is new to me)

James 5:19 & 20
My brethren, if any among you strays from the truth and one turns him back, let him know that he who turns a sinner from the error of his way will save his soul from death and will cover a multitude of sins. (you can’t turn them away from sin if you don’t tell them)

James 5:19 & 20
My brethren, if any among you strays from the truth and one turns him back, let him know that he who turns a sinner from the error of his way will save his soul from death and will cover a multitude of sins.
(you can’t turn them away from sin if you don’t tell them)

The Bible is very clear that we need to make judgments. We need to know God’s Word and be guided by the Holy Spirit so we will know what is wrong and right, and we need to point out sin in a person’s life and plead with them to repent of their sin and trust in Christ. At the same time, we must not be hypocritical in our judgment. We need to constantly examine our own lives and then “go and sin no more.”

What this all comes down to is, we are to judge sin where we see it. This judgement is not about the persons salvation, it’s about their sin. If I see my brother or sister sinning, out of love for that person, I will point it out and try and help. This is what it’s all about, and to help my brother or sister, I can’t be lost in the same sin, for if I didn’t overcome it, how can I help someone else? So I must remove the log (sin) from my eye (my life) before I can help them with that same sin.