Here is an example of a surjective morphism of sheaves $F\to G$ on $X$ such that for any open covering $\{ U_i\}_i$ of $X$, there is an index $i$ such that $F(U_i)\to G(U_i)$ is not surjective.
Let $X=\mathrm{Spec}(\mathbb C[t])$ be the affine line (say over $\mathbb C$) with origin $e$. Let $F=O_X$ be the structural sheaf of $X$. Let $R=O_{X,e}=\mathbb C[t]_{t\mathbb C[t]}$. Let $G$ be defined by $ G(U)= \left\lbrace\matrix{ 0 & \text{if } e\notin U \\ R & \text{if } e\in U.}\right.$ We can check that $G$ is actually a sheaf and $G_x=0$ if $x\ne e$ and $G_e=R$.
Consider the morphism $F\to G$ which on $U$ is the zero map if $e\notin U$ and is the localization map at $e$ otherwise. Then $F_x\to G_x$ is surjective for all $x\in X$. Hence $F\to G$ is surjective.
But for any open subset $U\ni e$, $F(U)\to G(U)=R$ is not surjective because if $U=D(f)$, then $F(U)=\mathbb C[t]_f$ is clearly strictly smaller than $R$.