By theorems of Wilking and Milnor (see On fundamental groups of manifolds of nonnegative curvature by Wilking), the fundamental group of a compact manifold with nonnegative scalar curvature is of polynomial growth. By Gromov's theorem, it is virtually nilpotent, hence amenable. Compact manifolds with amenable fundamental group have vanishing simplicial volume.

By theorems of Wilking and Milnor (see On fundamental groups of manifolds of nonnegative curvature by Wilking), the fundamental group of a compact manifold with nonnegative sectional (not scalar, I've misread the original question) curvature is of polynomial growth. By Gromov's theorem, it is virtually nilpotent, hence amenable. Compact manifolds with amenable fundamental group have vanishing simplicial volume.