The following definition is given in Hartshorne's book:
Definition: A subsheaf of a sheaf $\mathcal{F}$ is a sheaf $\mathcal{F'}$ such that for every open set $U \subseteq X$, $\mathcal{F'}(U)$ is a subgroup of $\mathcal{F}(U)$ and the restriction maps of the sheaf $\mathcal{F'}$ are induced by those of $\mathcal{F}$.
Now when Hartshorne says
"the restriction maps of the sheaf $\mathcal{F'}$ are induced by those of $\mathcal{F}$"
I'm sure he means that if we have open sets $V \subseteq U$ of $X$, then the restriction map for $\mathcal{F'}$ $$\rho'_{UV} : \mathcal{F'}(U) \to \mathcal{F'}(V)$$ is just the restriction of the restriction map for $\mathcal{F}$ $$\rho_{UV} : \mathcal{F}(U) \to \mathcal{F}(V)$$ to the subgroup $\mathcal{F'}(U)$. But for $\rho'_{UV}$ to even be well-defined $\mathcal{F'}(V)$ needs to be defined in such a way that we have $\rho_{UV}(\mathcal{F'}(U)) \subseteq \mathcal{F'}(V)$.
So the question is: Is this condition also implicit to the definition of a subsheaf?
If you think categorially in terms of what properties a subobject has in the category of sheaves of abelian groups I think that the answer would be yes to the above.