Here's an answer to your call for examples. Take $p$ to be a non-isolated closed point in your favorite topological space $X$. Let $H$ be a non-trivial abelian group.

## $Hom(F,G)_p\to Hom(F_p,G_p)$ fails to be surjective

Let $G$ be the constant sheaf $H$, and let $F$ be the skyscraper sheaf at $p$ with stalk $H$. Then $Hom(F,G)$ is the zero sheaf: any section of $F$ is trivial away from $p$, so a homomorphism would take it to a section of $G$ which is trivial away from $p$, but any section of $G$ which is trivial away from $p$ must be trivial, so every section of $F$ must be taken to the trivial section of $G$. So $Hom(F,G)_p=0$, but $Hom(F_p,G_p)=Hom(H,H)\neq 0$.

## $Hom(F,G)_p\to Hom(F_p,G_p)$ fails to be injective

Let $V=X\setminus p$, and let $F=G$ be the extension by zero of the constant sheaf $H$ on $V$ (i.e. it's the constant sheaf $H$ on $V$ and $F(U)=0$ if $U\not\subseteq V$). Then $F_p=G_p=0$, but $Hom(F,G)(U)$ contains a natural copy of $Hom(H,H)\neq 0$ for any $U$, so $Hom(F,G)_p$ contains a natural copy of $Hom(H,H)\neq 0$, so it's not zero.