Let $k$ be a field and $I$ and $J$ two sheaves $\mathcal{C}^{op}\to \mathrm{kVec}$ of $k$ vector spaces on a reasonable site $\mathcal{C}$ with terminal object $X$ which are injective and acyclic where the latter means $$ H^n(X, I)=0 \quad\quad\text{and}\quad\quad H^n(X,J)=0 $$ for all $n\neq 0$. Can anything be said about the groups $$ H^n(X, I\otimes_k J) \quad? $$ I particular I wonder, if $I\otimes_k J$ still acyclic.
(I have some information about the cohomology of tensor products $F\otimes_{\mathcal{O}_X}G$ of $\mathcal{O}_X$-modules, but here, the tensor product is over the field $k$. As the functor $-\otimes_k V$ is exact for $k$ vector spaces, one can probably say more which I am missing.)