One might think on the first reading of this proverb that God is encouraging those who have hatred to let their mouths voice it openly. But you need to remember that many of the proverbs start with a basic statement - and then that statement is explained with an added comment that further defines it. That is the way to understand the proverbs. If we do not read them this way - it is easy to grab a partial proverb and make it say something it was never meant to say.
What we have here is a proverb about hidden hatred later escaping through slander. The first part of this proverb speaks of one who is concealing their hatred by saying things they do not believe. An example of this is someone who says nice things or supportive things of someone when they are around that person. Inwardly they despise this person - but they do not allow their hatred to be expressed when around them. But do not be deceived by their lying lips - because deep within them they are churning with a hatred that will surface in time.
Their hatred surfaces in what is said in the second part of this proverb. They begin to spread slander. They will not speak openly about their hatred of a person, a viewpoint, a decision, etc. But later they will begin a slander campaign to express their hatred. Put very plainly - they won't tell someone openly they hate them - but in private and with others they will let their hatred fly by slandering them often.
The one who does this is a fool. The Scriptures says that it is better to openly rebuke someone than to secretly slander them. If there is a biblical issue that needs to be addressed, then address it privately with the person. If there is a view that needs to be opposed - then oppose it where a debate of ideas can be had. If there is a decision that is wrong - then openly state that it is wrong. To allow a hatred to build up in our hearts is not just foolish - it is dangerous. In time it will surface in slanderous gossip. We WILL eventually have our hatred come out - and when it does - it will not be pretty.
You may not like someone who openly confronts you - who rebukes and questions a view you have taken - or who disagrees with you publically. But one thing is for sure. You have far more respect for someone who does this, then for someone who speaks open words of support - but follows them up with slanderous gossip when you are no longer around. Such a man is a fool - and his actions will come back to roost eventually. Speak the truth in love - and if you disagree with someone and do not feel God wants you to speak out - then love them by interceding for them. Hatred, gossip, and slander has NEVER helped any situation where it was used. It has only made for fools and for deep hurt by those who have been injured by it.