Ultimately, it depends on the sparsity of $A$ and $B$ and the symmetry of the resulting hadamard product.
The hadamard product will aggregate the sparsity structure of $A$ and $B$. So if one or the other is sparse, the product is also sparse (and may be more so if $A$ and $B$ have difference sparsity structures). So any sparse direct solver would be appropriate. However, if the matrix is extremely large, memory becomes an issue and sparse iterative methods are just about the only choice.
Even if $A$ and $B$ are dense, the hadamard product is an $O(n^2)$ operation, which is insignificant compared to a direct linear solve of $O(n^3)$ operations. So even if A and B are extremely large dense matrices, simply computing the hadamard product before solving the resulting system is not that much more work compare to solving the system.
In the dense case, the key question becomes whether the resulting matrix is hermitian or not. In the worst case (non-hermitian hadamard product), gaussian elimination is pretty much the only thing you can do. If it is hermitian, a Cholesky (definite ) or Bunch-Kaufman (indefinite) algorithm would be appropriate.