Does positive scalar curvature imply vanishing of the simplicial volume on a closed Riemannian manifold? Are there any relationship between the scalar curvature and the simplicial volume? 
The simplicial volume is zero (positive) on Torus (Hyperbolic manifold) and those manifolds does not admit a Riemannian metric with positive scalar curvature.  What do we know  about the simplicial volume of a Riemannian manifold with positive scalar curvature?
 A: In a preliminary version of what would become Gromov's "A Dozen Problems, Questions and Conjectures about Positive Scalar Curvature", he writes on page 88:

Neither is one able to prove (or disprove) that manifolds with positive scalar
  curvatures have zero simplicial volumes.
  Possibly, these conjectures need significant modifications to become realistic.

Simplicial volume didn't make it into the published version of these notes, maybe because he thought his conjectural relationship between scalar curvature and simplicial volume was too hopeless to be worth mentioning, but at any rate this is reasonable evidence that this question was an open problem in 2017, and I haven't seen any evidence of progress since then.
A: 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.
