Let $X\rightarrow \operatorname{Spec}A$ be a morphism of schemes, $I$ an ideal of $A$ and $\mathcal{F}$ an $O_X-$module. The morphism gives a morphism $A\rightarrow O_X(X)$ and so for any open subset $U\subset X$, we've got a morphism $A\rightarrow O_X(U)$ and so $\mathcal{F}(U)$ can be equipped with an $A-$module structure and so we can consider the $O_X-$module $I\mathcal{F}:U\mapsto I\mathcal{F}(U)$ which is a submodule of $\mathcal{F}$.
I want to show that $IH^n(X,\mathcal{F}/I\mathcal{F})$ where $H^n$ denotes sheaf cohomology. I'd like to check whether my answer is correct, and I have a feeling I'm complicating things so if anyone has an easier solution I would appreciate it!
For any $x\in X$, $(\mathcal{F}/I\mathcal{F})_x$ is an $O_{X,x}-$module and as an $A-$module it is annihilated by $I$ so $(\mathcal{F}/I\mathcal{F})_x$ is an $A/I-$module.
Let $(\mathcal{F}/I\mathcal{F})_x\rightarrow J_x$ be a monomorphism to an injective $A/I-$module. Now we consider the sheaf of abelian groups $\mathcal{J}$ defined by $\mathcal{J}(U)=\prod_{x\in U}J_x.$ We can check that this is an injective sheaf of abelian groups and that the canonical morphism $\mathcal{F}/I\mathcal{F}\rightarrow \mathcal{J}$ is injective. Each $\mathcal{J}(U)$ is an $A/I-$module so we can construct an injective resolution $(\mathcal{J}^k)$ of $\mathcal{F}/I\mathcal{F}$ in the category of sheaves of abelian groups such that $\mathcal{J}^k(U)$ is an $A/I-$module for all $k$ and $U\subset X.$ So the cohomology $H^n(X,\mathcal{F}/I\mathcal{F})$ of the complex $(\mathcal{J}^k(X))$ is an $A/I-$module so $IH^n(X,\mathcal{F}/I\mathcal{F})=0.$
Thank you for your answers!
This might be equivalent to what you did if you go through all the proofs (and indeed you have understood the point, which is that an $A/I$-module is just an $A$-module killed by $I$). But it might still be helpful for you. Let us write down the setup in fancier symbols. We have a morphism of schemes $f: X \to \mathrm{Spec} A$, and a quasicoherent sheaf $\mathcal{F}$ on $X$. We also have a closed embedding $\iota: \mathrm{Spec}(A/I) \to \mathrm{Spec} A$. Let $$\iota': X \times_A A/I \to X$$ be the closed embedding induced by base-change to $X$ via $f$ (this is basically the preimage in $X$ of the closed subscheme cut out by $I$). Note that $\iota'$ is an affine morphism. The definition of the sheaf $\mathcal{F}/I\mathcal{F}$ considered as a quasicoherent sheaf on $X$ is $$\mathcal{F}/I\mathcal{F} := (\iota')_*(\iota')^*\mathcal{F}$$ (the pullback quotients by $I$ and the pushforward restricts scalars from $A/I$ to $A$ in order to consider what is naturally a sheaf on the closed subscheme $X \times_A A/I$ as a sheaf on all of $X$). Now we may apply the fact that $$H^n(X, \mathcal{F}/I\mathcal{F}) = H^n(X, (\iota')_*(\iota')^*\mathcal{F}) \cong H^n(X \times_A A/I, (\iota')^*\mathcal{F})$$ as $A$-modules (use e.g. Vakil 18.1(v)). In particular, the guy on the right is an $A/I$-module (which is where the $A$-module structure comes from), so we conclude that the $A$-module on the left is killed by $I$, as desired.