Here is a counter example in the general case:
Take $E = [1] = 0 \to 1 $. $D = \{0\} \coprod \{1\} $ and $C = \{1\}$. where all maps are weak equivalence.
The lax-pullback is $\{id:1 \to 1\}$, and the localization of $E$ and $C$ are both the terminal category, while the localization of $D$ is $D$ itself. so the pullback of the localization has two non isomorphic objects, while the pullback only has one objects.
A special case of that question that is well studied is when all maps are weak equivalences and one take a pseudo-pullback. This corresponds to whether a homotopy pullback in the Joyal Model structure is preserved by the localization corresponding to Kan-Quillen model structure.
This is not the case in general, in fact, it is not very often the case. Typical condition under which this is true are given by Quillen's Theorem B, or its generalization to infinity category ( and directly stated in terms of pullback square) which you can find as theorem 4.6.11 of Cisinski "Higher categories and homotopical algebra" (see condition (iii)). One can deduce result about lax pullback from this as well.
I would expect a similar result for general localization might be possible, but I don't think I have even seen it.