The Bible says "Thou shall not judge, you Hypocrite!" now it doesn't just say that you shouldn't judge. It says that you shouldn't be a hypocritical judge.
So, yes and no. Being a hypocritical judge is something that the Bible is against, however judging someone else in a loving/kind manner is something that's okay. Especially if it's to help the person, not bring them down.
In pursuit of full disclosure I will admit that I am an agnostic, but I will use the Bible's own logic to refute this point. Nowhere in the bible does it say make sure you go around and tell others in what ways they aren't following the word of god. It does however, say that you should inform people of the good news of the lord. Therefore if you are criticizing another for their actions you aren't really informing them of the good news of god. If you were you'd have no reason to mention the sin at all. The reason you bring it up is because it is a behavior that goes against your personal code of ethics. All you are really attempting to do is apply your own morality onto others which despite how religion has played out throughout history, was not what the bible told people to do. If you were truly worried about the other "burning in fiery damnation," you would simply inform them about the good news of god and leave it up to them about what to do with the information. To do otherwise is ultimately hypocritical.
I'm speaking from a Christian perspective.
It is hypocritical for anyone to JUDGE another for sinning, for then one merely judges oneself because no one is sinless (except, to Christians, Jesus Christ). Thus, it is sinful to judge others (that is, hold them in contempt or to presume to condemn them for some sin or another [eternally and personally speaking. I am not talking about a judge judging a criminal for example.]).
However, to criticize another is different. Just as to criticize someone's cooking is in the hopes of making them a better cook, so criticizing the sin of others is a way to help them be a better person.
Obviously one has to be very careful to make sure that one is not judging, and the line can be very thin. But there is definitely a difference between casting judgement on someone (either privately or publicly) and criticizing them personally. That is to say, accepting that you yourself sin, informing another person of their sin, and then telling that person how they can amend their sin (in a way that you are also amending for your sin).
Christians do it the world over in confession and absolution.
It is not a very good moral construction designed to promote personal responsibility. Firstly, the very idea of being held accountable for anything your ancestors did is outrageous. Secondly, being able to transfer all of one's faults onto an imaginary scapegoat in order to escape punishment is simply deceitful and immoral. Being able to forgive yourself is great, but we have a social responsibility to identify everyone's shortcomings and accept the consequences ourselves.
Somehow I think God would be perfectly fine with anyone who would criticize the sins of Ted Bundy, Burnie Madoff or Hitler. Sin needs to be brought into the light in order to be exposed for the filth it is. While I agree that criticizing others for mild sins is usual hypocritical, calling out major sin is a worthy cause.
Not if you are merely, as the Bible says "speaking the truth in love." You are then only, as it is also said, "saving them from the flame." It is up to you to do this, but up to them to respond. Do force them to change, but make the statement and approach them like "Hey man, I believe what you are doing is wrong." If they are not saved yet, then show them the Gospel. From there, just be a good influence and help them in making positive steps.
It definitely isn't a good thing to criticize someone for their bad deeds. Instead, show them their mistkaes and how ro improve. Tell them politely what they did wrong, and how to amend. Infact, helping others improve their deeds, will help your chances of getting into Heaven when you pass.