meself asks: I never heard that "saying", and still haven't, outside the song. Is "Little pitchers have big ears" a saying that anyone else is familiar with, outside the song?

Absolutely. In fact, the saying is so familiar that, it seems to me, one rarely hears it completed. The usual way it's used is to say "Little pitchers" and point -- often with the head rather than the hand, because it's less obvious that way.

Looking at cnd's list, I wonder if it might not be something of a midwesternism (I'm in Minnesota, and my family came from Michigan). Known outside the American Midwest, of course, but perhaps more common here.