This question says really screams "I don't know how to compute nef cones and find interesting birational morphisms".
If $f:X \to Y$ is a birational morphism between smooth $X,Y$ and an algebraic fiber space, is it possible there is a ray $R \in NE(X)$ s.t $K_X \cdot R \geq 0$?
I'm asking whether Mori theorem tells us "everything" there is to know about morphisms.
Whoever asked the question doesn't even try the simple examples.
Start with $Y = \mathbb{P}^2$ and blowup at a sequence of points to get $X$.
If we blow up $p_1$ to get $E_1$ and then $p_2$ on $E_1$ with exceptional divisor $E_2$ we'll get (meaning strict transforms here) $E_1^2 = -2, E_2^2 =-1$, and $K_X = f^*(K_Y) + E_1 + 2E_2$ and thus $E_1$ has $K_X \cdot E_1 = 0$.
If you want positive intersection, then blowup again on $E_1$ to get $f^*(K_Y) + E_1 +2E_2 + 2E_3$, now $E_1^2 =-3$ but still the intersection $K_X \cdot E_1$ is positive