Under which conditions do the restriction maps of sheaves of modules occur to be injective? Let $(X,\mathcal{O}_X)$ be a scheme which is at least integral. Further let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules. Now under which conditions on $\mathcal{F}$ and/or $(X,\mathcal{O}_X)$ do we obtain that the restriction maps 
$$\rho_{U \subseteq V}: \mathcal{F}(V) \rightarrow \mathcal{F}(U)$$ 
for open subsets $U \subseteq V$ of $X$ are injective?
In the case $\mathcal{F} = \mathcal{O}_X$ we need that $(X,\mathcal{O}_X)$ is integral (hence the above assumption) which proves that such sheaves indeed exist. Are we able to classify such sheaves by certain conditions? Maybe some property like quasi-coherence or $\mathcal{O}_X$-modules which are embedded in the function field of $X$.
I'll be grateful for any kind of input.
 A: Consider the case when $X$ is affine, $X=Spec(A)$ and $M$ is a coherent module, that is an $A$-module, the question is equivalent to the following:
Let $f\in A$, and $g:M\rightarrow M_f$ the canonical map, where $M_f$ is the localization of $M$, is $g$ injective ? This is true if $M$ does not have torsion.
A: To complement the other answer, this is how you reduce to the affine case:
Let $X = \bigcup_{i \in I} U_i$ be an affine open cover, $V \subset X$ any open subset and $f \in \mathcal F(X)$ with $f_{|V}=0$. For the sake of notation let $V_i := U_i \cap V$.
In particular $f_{|V_i}=0$ for all $i$. By the result in the affine case, we have an injection $\mathcal F(U_i) \to \mathcal F(V_i)$, hence $f_{|U_i}=0$ for all $i$. The sheaf properties yield $f=0$, the desired result.
A: See the following picture from my personal notes. For integral scheme $X$ and quasi-coherent modules $\mathcal{F}$ (so we can check this in the algebra setting),

the restriction maps are injective if and only if $\mathcal{F}$ is
torsion free.

In this case we also have a nice characterization of $\mathcal{F}(U)$ for any non-empty open $U$ (not just affine open),
$$\mathcal{F}(U)=\bigcap_{p\in U}\mathcal{F}_p$$


