Let $X$ be the complex manifold $\mathbb{C}^* = \mathbb{C}\setminus\{0\}$ with its usual structure sheaf. For every integer $n$, let $e_n:U_n\to X$ be the open inclusion of the submanifold $\mathbb{C} \setminus [0,n]$, i.e., the complement of the interval $[0,n]$ in the nonnegative real axis. Let $e:U\to X$ be the open inclusion of $\mathbb{C}\setminus [0,\infty)$, i.e., the complement of the nonnegative real axis. For every integer $n$, define $\mathcal{F}_n = (e_n)_*\mathcal{O}_{U_n}$. Define $\mathcal{F}=e_*\mathcal{O}_U$. Via the sequence of inclusions $U_1\supset U_2 \supset U_3 \supset \dots\supset U$, there is a sequence of restriction homomorphisms $\phi_{n,n+1}:\mathcal{F}_n \to \mathcal{F}_{n+1}$ compatible with the restrictions $\psi_n:\mathcal{F}_n\to \mathcal{F}$.
Since stalks commute with colimits, for every $p\in U$, $\varinjlim(\mathcal{F}_n)_p = \mathcal{O}_{U,p}=\mathcal{F}_p$. Also, for every $p\in [0,\infty)$, for every $n>p$, $(\mathcal{F}_n)_p\to \mathcal{F}_p$ is an isomorphism. Thus, checking on stalks, the sequence of homomorphisms $(\psi_n:\mathcal{F}_n\to\mathcal{F})$ is the direct limit of the sequence $(\mathcal{F}_n,\phi_{n,n+1})$. On the other hand, $\log(z)$ is a global section of $\mathcal{F}$ that is not a global section of any $\mathcal{F}_n$.