Building off of the nice answer of Iosif Pinelis, and Wojowu's comment on it, we have:
Observation: A space $X$ fails to have the OP's property if some nonempty open $U \subseteq X$ is second countable.
(Proof: Let $(U_n)$ enumerate a basis for $U$.)
In the other direction, let me observe that there is at least one nice(ish) properties that do imply the OP's property. But, by the observation above, any such property must contradict metrizability (some kind of "bigness" property). Recall that a space is ccc if it does not admit an uncountable family of pairwise disjoint open subsets.
Observation: A space $X$ has the OP's property if no nonempty open $U \subseteq X$ is ccc.
Proof: Assume that no nonempty open $U \subseteq X$ is ccc, and let $(U_n)$ be a sequence of nonempty open subsets of $X$. We construct the $V_n$ by recursion. First, fix an uncountable family $\mathcal U_0$ of pairwise disjoint nonempty open subsets of $U_0$. All but countably many $V \in \mathcal U_0$ have the property that $U_n \not\subseteq \overline V$ for all $n$. Let $V_0$ be some such $V \in \mathcal U_0$. Next, fix an uncountable family $\mathcal U_1$ of pairwise disjoint nonempty open subsets of $U_1 \setminus \overline V_0$. All but countably many $V \in \mathcal U_1$ have the property that $U_n \not\subseteq \overline V$ for all $n$. Let $V_1$ be some such $V \in \mathcal U_1$. Generally, at step $k$ fix an uncountable family $\mathcal U_k$ of pairwise disjoint nonempty open subsets of $U_k \setminus (\overline V_0 \cup \dots \cup \overline V_{k-1})$. All but countably many $V \in \mathcal U_k$ have the property that $U_n \not\subseteq \overline V$ for all $n$. Let $V_k$ be some such $V \in \mathcal U_k$. QED
One example of a space with this property is $\mathbb N^*$, the Stone-Cech remainder of the countable discrete space $\mathbb N$. (I suppose one could argue this is not a very "nice" space, but one would be wrong. :))