What is an interesting example of that? Things like $Spec(K) \to Spec(L)$ do not count cause they are not interesting.
Dear Dung, a pleasantly geometric example of a quasi-finite, separated, but not finite morphism is the projection of the hyperbola $xy=1$ in the affine $x,y$ plane on the $x$-axis. Its image is the affine line minus the origin. It is clearly quasi-finite (even injective) but not finite, since its image is not closed . (Also, morphisms between affine schemes are separated)