Let $H$ be a Hilbert space, and \begin{equation} D\colon \text{dom}(D)\to H \end{equation} be a closed symmetric linear operator, and $\text{dom}(D)\subset H$ is dense.
Now we have a sequence $x_n\to x$ in $H$ with $x_n\in\text{dom}(D),x\in\text{dom}(D^*)$ and $Dx_n\rightharpoonup D^*x$, then can we say $x\in\text{dom}(D)$?
Yes. The point $(x,D^\ast x)$ lies in the weak closure of the graph of $D$. By the Hahn-Banach theorem, the weak and strong closure of convex sets (in particular, subspaces) coincide. Thus $(x,D^\ast x)$ lies in the strong closure of the graph of $D$, which is just the graph of $D$ since $D$ is closed.
In the case of Hilbert spaces you don't even need the Hahn-Banach theorem, but can use the (more elementary?) Banach-Saks theorem: If $y_n\rightharpoonup y$, then there exists a subsequence $(y_{n_k})$ of $(y_n) $ such that $$ z_N=\frac 1 N\sum_{k=1}^N y_{n_k} $$ converges to $y$ strongly.