any known universality results of random matrices with non-independent entries?

Of course there is:

For example: this paper by Lubinsky describes universality for a special class of tridiginal matrices. For example the condition off-diagonals $equiv 1$ and entries on the diagonal are in $ell^1(mathbb{Z})$ would suffice. So the matrices are
$$
H_N = begin{pmatrix} b_1 & 1 & \
1 & b_2 & 1 & \
& 1 & b_3 &1 & \
& & ddots & ddots & ddots \
& & & 1 & b_N end{pmatrix}
$$
with $sum_{n=1}^{infty} |b_n| < infty$.

This can be further generalized see: Avila--Last--Simon. Of course all these results are for special tridiagonal matrices (Jacobi operators).

Last, there is also the work by Deift et al. See the book.