Comment by Gro-Tsen on Smooth analogue of Cartan's Theorem B

My memory may serve me incorrectly, but if I recall correctly, the real-valued $C^\infty$ functions on a (paracompact) $C^\infty$ manifold $X$ form a “fine” sheaf $\mathscr{F}$ (“fine” means any section on a closed set can be extended to a section on all of $X$), so it is acyclic: $H^p(X,\mathscr{F}) = 0$ for $p>0$. For the constant sheaf $\mathbb{R}$, on the other hand, obviously, $H^p(X,\mathbb{R}) = 0$ would hold only with extra assumptions, contractibility being a sufficient one. Which of these two situations are you more after?