I’ll generalize Dylan’s example. If $\sigma = \langle x_n:n\in\omega\rangle$ is a convergent sequence of distinct points in a $T_1$-space $X$, the sets $A_n = \{x_n\}$ are an example. Because $X$ is $T_1$, singleton sets are closed, and the limit of the sequence clearly belongs to $\left(\operatorname{cl}\bigcup\limits_{n\in\omega}A_n\right)\setminus\bigcup\limits_{n\in\omega}A_n$.
In fact, all you need is a space $X$ with a non-isolated point $x$ of countable pseudocharacter, meaning one that is the intersection of countably many open sets: if $\{x\} = \bigcap\limits_{n\in\omega}V_n$, where each $V_n$ is open, just let $A_n = X \setminus V_n$ for each $n\in\omega$. Clearly the $A_n$ are closed and their union is $X\setminus \{x\}$; but $x$ isn’t isolated, so is must be a limit point of $\bigcup\limits_{n\in\omega}A_n$.
If the family of sets needn’t be countable, any non-discrete $T_1$-space works. Let $X$ be such a space, and let $x$ be a non-isolated point of $X$. For each $y\in X \setminus \{x\}$ let $A_y = \{y\}$. As in my first example, each $A_y$ is closed, but the union of the $A_y$ is $X\setminus \{x\}$, which clearly has the non-isolated point $x$ in its closure.