Does proclaiming that all non-believers won't be saved count as passing judgement? Romans 10:9 (KJV)
  That if thou shalt confess with thy mouth the Lord Jesus, and shalt believe in thine heart that God hath raised him from the dead, thou
  shalt be saved.


If I interpret Romans 10:9 as saying all those who don't confess with their mouth and don't believe in their heart, that they will not be saved and I proclaim this; am I therefore sinning by passing judgement on those who do not confess or believe?


  Romans 2:1 (KJV)
  Therefore thou art inexcusable, O man, whosoever thou art that judgest: for wherein thou judgest another, thou condemnest thyself;
  for thou that judgest doest the same things.
There is a difference between condemning and being discerning. One is using the good judgement God gave you, the other is judging in the place of God. 

It should be clear that Christians are supposed to be "as wise as serpents but as gentle as lambs. ". Throughout the NT, we are warned to avoid false teachers and cast them out of our midst. We are to encourage and rebuke our brothers  

But note the context- in the cases where we are to use discernment, it is in a personal, relating body of believers.  We are not called to be God's policemen, forcing a system of morality on unbelievers. Rather, it is a privilege to be subject to the discipline of God - for after all, he only disciplines those he loves. 

If a brother is calling himself a Christian, he invites discipline, rebuke, and encouragement. He sees it as a way to be the person God created him to be. If not, there is no use in casting your pearls before swine.