Matthew 7: 1 “Do not judge, or you too will be judged. 2 For in the same way you judge others, you will be judged, and with the measure you use, it will be measured to you. 3 “Why do you look at the speck of sawdust in your brother’s eye and pay no attention to the plank in your own eye? 4 How can you say to your brother, ‘Let me take the speck out of your eye,’ when all the time there is a plank in your own eye? 5 You hypocrite, first take the plank out of your own eye, and then you will see clearly to remove the speck from your brother’s eye.

The world often takes this passage of the Bible (out of its context) and uses it to accuse Christians of being “judgmental” when they speak of sin. The context of this passage from Matthew is revealed in verse 5 - the Christian is not to judge hypocritically or self-righteously.

Scripture repeatedly exhorts believers to evaluate carefully (John 7:24) and to choose between good and bad people and things:
~ sexually immorality (1 Corinthians 5:9)
~ those who masquerade as angels of light (2 Corinthians 11:14)
~ dogs, those who are evil (Philippians 3:2)
~ false prophets (1 John 4:1)
The Christian is to “test everything” (1 Thessalonians 5:21).

However, while Jesus is telling His disciples not to judge one another hypocritically, He also says…

“Do not judge according to appearance, but judge with righteous judgment.”

(John 7:24)

Thus, if a man steals, lies, commits adultery or murder, etc., the Christian can make a (righteous) moral judgment and say that the actions were morally wrong, and that these sins will have eternal consequences. This is stating (teaching, repeating) the Moral Law already revealed by God - God’s own judgement of what is right or wrong.

To say of oneself in relation to Christians, in a manner of accusing them of being judgemental for stating God’s moral laws, something like “I will not judge but will leave it up to God to judge what is right and wrong” simply ignores what Jesus has told Christians to do, and what God has already revealed.

1 Corinthians 2: 12 We have not received the spirit of the world but the Spirit who is from God, that we may understand what God has freely given us. 13 This is what we speak, not in words taught us by human wisdom but in words taught by the Spirit, expressing spiritual truths in spiritual words. 14 The man without the Spirit does not accept the things that come from the Spirit of God, for they are foolishness to him, and he cannot understand them, because they are spiritually discerned. 15 The spiritual man makes judgments about all things, but he himself is not subject to any man’s judgment.

As explained by Visiblesoul*…

The spiritual man judges all things but he does not judge them by his own understanding or “human wisdom”; he judges all things by the revealed knowledge of God. He is not subject to any man's judgment because the natural man cannot comprehend why the spiritual man thinks and acts the way he does. But the spiritual man does understand the natural man because the spiritual man once lived in the same natural mindset. As a wise man once said —

The path of the righteous is like the first gleam of dawn,
shining ever brighter till the full light of day.

But the way of the wicked is like deep darkness;
they do not know what makes them stumble.
(Proverbs 4:18-19 NIV)

*(see a previous post to Judah’s Journal on this subject: Judgement - and please don't shoot the messenger!)

For a more detailed treatment of this subject, read:
“Who Are You to Judge Others?” - In Defense of Making Moral Judgments by Paul Copan

It's been said that the most frequently quoted Bible verse is no longer John 3:16 but Matthew 7:1: “Do not judge, or you too will be judged.” We cannot glibly quote this, though, without understanding what Jesus meant. When Jesus condemned judging, he wasn't at all implying we should never make judgments about anyone. After all, a few verses later, Jesus himself calls certain people “pigs” and “dogs” (Matt. 7:6) and “wolves in sheep's clothing” (7:15)! Any act of church discipline (1 Cor. 5:5) and rebuking false prophets (1 John 4:1) requires judgment. What Jesus condemns is a critical and judgmental spirit, an unholy sense of moral superiority. Jesus commanded us to examine ourselves first for the problems we so easily see in others. Only then can we help remove the speck in another's eye—which, incidentally, assumes that a problem exists and must be confronted.

Check out what John (Omnipotent Grace) has to add in his post here.
As John goes on to explain concerning the misuse of Matthew 7:1 …

Of course, closer examination of the passage and the verse demonstrate that to use it in such a manner is quite ridiculous. We are all to be judged one day on the basis of our life on earth.

But if we have to believe those who so love to quote this verse and use it as ammunition for whatever heresy they are defending, then it is quite simple:

If we don't judge, then we won't be judged, and therefore, if we are not judged, we won't have to account for our sins. And if we are not called to account for our sins, we have no need for a Savior, and we are assured of our safe entry into the presence of God. And that, dear friend, is heresy of the highest order.

So quite simply, taking the argument to its logical conclusion, not judging people is replacing Christ.