Regarding schemes: on a scheme $X$, the functor of global sections is exact iff no quasicoherent sheaves have higher cohomology. By <a href="http://amathew.wordpress.com/2012/08/01/serres-criterion-for-affineness-as-morita-theory/">Serre's criterion for affineness</a>, for reasonable schemes this is equivalent to $X$ being affine, essentially because in this case global sections is a functor $\text{QCoh}(X) \to \Gamma(X, \mathcal{O}_X)\text{-Mod}$ and exactness is one of the only obstructions to it being an equivalence.