I came up with the following example and I wanted to know if it is correct. Let $\mathcal F$ and $\mathcal G$ be sheaves of smooth functions on $\mathbb R$. Consider the smooth function $f$ constructed in the following manner (I think the heart of the matter is if my $f$ is indeed smooth).
First, let $b$ be a bump function vanishing on $[-1,0]$, and $g$ be any smooth function vanishing on the closed set {$1/n$}. Then define $$f = bg^{-1}.$$
(My only problem in checking that this is a smooth function is that I know that g is not regular at $0$, i.e. it doesn't vanish with multiplicity 1. Thus I don't know if multiplying by my bump function will be enough to make f vanish at zero.)
Now consider the map in $\mathcal {Hom (F_0,G_0)}$ given by multiplication by $f$. It is clear that this map cannot lie in $\mathcal {Hom (F,G)}_0$, for there is no nbhd around $0$ s.t. whatever multiplied by $f$ is smooth in it.