Here is a counter exemple:
Take $E = [1] = 0 \to 1 $. $C = \{0\} \coprod \{1\} $ and $D = \{0\}$. where all maps are weak equivalence.
The (pseudo-)pullback is $\{0\}$, and the localization of $E$ and $D$ are both the terminal category, while the localization of $C$ is $C$ itself. so the pullback of the localization has two non isomorphic objects, while the pullback only has one objects.
Assuming they are model categories doesn't really help. Unless one of the two functor have some very good properties. For example, that it can lift weak-equivalence/fibrations factorization.
More generally, a special case of your question is when all maps are weak equivalence. 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)).
I would exepect a similar result for general localization might be possible, but I don't think I have even seen it.