Here is the minimal example I believe (which I found an hour after asking the question).
Let X be space with 3 points, ordered like this: 1 < 3, 2 < 3, with the order topology: a set U is open if x in U, x < y, implies y in U.
Then the only projective sheaves of abelian groups are of the form Z_{\ge x} and their direct sums. And the constant sheaf Z is not a quotient of any of these.
I think one classifies projective sheaves on any finite topological space just as easily.
VA
P.S. Jon Woolf's example is very nice.