In [Hartshorne, III.3] he proves that injective modules over $R$ give flasque sheaves over $Spec\ R$. I presume that's because they don't give injective sheaves, and flasque is the consolation prize. Is there an easy counterexample?

EDIT: in III.3 he's assuming Noetherian. And he's already proved in II.5.5 the equivalence of categories of $R$-modules and quasicoherent ${\mathcal O}_{Spec\ R}$-modules. (And that injective sheaves are flasque, in III.2.)

EDIT: his proof that injectives are flasque uses some non-quasicoherent sheaves. So the ingredients "injective R-modules give injective objects in the category of quasicoherent sheaves [II.5.5]" plus "injective objects in the category of sheaves are flasque [III.2]" isn't enough for the result he gets in III.3, that injective R-modules give flasque sheaves.