Consider a coherent sheaf $F$ over a locally noetherian scheme $X$. Further assume that $F$ has a global section $s:\mathcal{O}_X\to F$. Then, $J:=\ker s$ is an ideal sheaf defining a closed subscheme of $X$ and there is an injective map $s':\mathcal{O}_X/J\hookrightarrow F$. Let $I$ be the ideal sheaf defining the support of $F$.
Since the section may vanish in more points than the support we find that $I\subseteq J$. I want to know what can we say about the sheaf $\mathcal{O}_X/I$ with respect to $\mathcal{O}_X/J$ and $F$.
1) Do we have an inclusion $\mathcal{O}_X/I\hookrightarrow F$ ?
2) Is there a more concrete relation between $\mathcal{O}_X/I$ and $\mathcal{O}_X/J$ ?
Since this problem is essentially local in nature, I'll restate it in terms of commutative algebra. I apologise if I made a mistake somewhere, or if this is all already obvious to you.
So let's pick an open affine subscheme ${\rm Spec \ }A \subset X$, where $A$ is a noetherian ring. On this open affine, the coherent sheaf $\mathcal F$ is of the form $\widetilde M$ for some finitely-generated $A$-module $M$. It is well-known that the support of $\widetilde M$ is the closed subscheme associated to the annihilator ideal ${\rm Ann \ }M \subset A$. [Hartshorne II Exercise 5.6(b).]
Now let's consider the morphism $s: \mathcal O_X \to \mathcal F$. On our open affine subscheme, this corresponds to a morphism $A \to M$ of $A$-modules. Such a morphism is uniquely specified by chosing an element $m \in M$ to be the image of $1 \in A$; the morphism $A \to M$ is then given by $r \mapsto rm$. Therefore, the kernel of this morphism is the ideal ${\rm Ann \ }m \subset A$.
So what can we conclude?
There is a natural surjection $A / ({\rm Ann \ }M) \twoheadrightarrow A / ({\rm Ann \ } m)$, corresponding locally to a surjective sheaf morphism $\mathcal O_X / I \twoheadrightarrow \mathcal O_X / J$.
There is a natural injection $A / ({\rm Ann \ }m) \hookrightarrow M$, corresponding locally to an injective sheaf morphism $\mathcal O_X / J \hookrightarrow\mathcal F$.
Composing these two morphisms, we get a morphism $A/({\rm Ann \ }M )\to M$ sending $[r] \mapsto rm$, which corresponds locally to a sheaf morphism $\mathcal O_X / I \to \mathcal F$. But this morphism is not injective on this open affine subscheme unless ${\rm Ann \ }M = {\rm Ann \ }m$, and this morphism is not surjective on this open affine subscheme unless $M = Rm$. Globally, the equivalent statement is that the morphism is not injective unless the closed subschemes associated to $I$ and $J$ are isomorphic, and the sheaf morphism is not surjective unless $\mathcal F$ is globally generated by the section $s$.
And quite honestly, I don't think we can say much more. But there are plenty of users on this site who are more experienced than me, so don't take my word for it.