In Act I Scene 3, what does "give thy thoughts no tongue" mean?
Polonius' advice to his son, when considered on the whole, seems to place a heavy emphasis on protecting one's reputation. He starts his advice with "And these few precepts in thy memory / Look thou character." He wants Laertes to be very aware of his actions and what others may know or say about him that could be judged by his speech or actions. In three separate places his tells Laertes to not speak his mind. First in the line you asked about, and later when he says, "Give every man thine ear, but few thy voice" and lastly the next line says, "Take each man's censure, but reserve thy judgement."
Polonius has an important position in the court and doesn't want either of his children's actions to reflect badly on the family name. This speech and the one that follows to Ophelia show Polonius' emphasis on ettiquette and appearances.
The line that you mention, taken just on its own, means something like "do not say what you are thinking." To "give tongue" to something means to speak it out loud -- to use your tongue to express it.
What is going on in this scene is that Polonius is giving advice to his son, Laertes, before Laertes goes to Paris. A lot of his advice is pretty stereotypical, obvious stuff like this. In the part that you are asking about, he tells Laertes that he should not let other people know what he is thinking and he should not act too hastily on what he thinks either.