The first one that comes to my mind from my English classes years ago, is:
Make new friends,
but keep the old;
one is silver,
the other is gold.
I also found this youtube video which is related. (:
There are many such idioms, not so many proverbs that I can think of.
Maybe the most famous proverb on keeping silent is from Proverbs 17:28:
Even a fool, when he holds his peace, is counted wise: and he that shuts his lips is esteemed a man of understanding.
It's repeated in various parts of Scripture in different phrasing. I think it's the basis of the familiar
It is better to be silent and be thought a fool, than to speak and remove all doubt.
The following is attributed to Confucius:
Silence is a true friend who never betrays.
In addition to the idioms @Dan Bron mentions, there is also flying under the radar, originally meaning avoiding detection, but now meaning avoiding negative attention as well.
However, even wise people often embrace silence, and there is a saying in teaching that no question is a stupid question. How can one learn if one doesn't ask questions?
Best Answer
Wolfgang Mieder, A Dictionary of American Proverbs (1992) lists three sayings that field researchers collected in various parts of the United States:
Rosalind Fregusson, The Facts on File Dictionary of Proverbs (1983) renders the Bacon proverb as follows:
Of course, if it first appeared in the Novum Organum, Bacon must have written it in Latin.
Fergusson lists a number of other proverbs about proverbs, but without specifying their place of origin. My two favorites from her collection are
and
This last saying goes back at least as far as James Kelly, A Complete Collection of Scottish Proverbs: Explained and Made Intelligible to the English Reader (1721). Kelly explains its origin as follows: