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.

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.

Another situation where a positive answer in this direction is possible, is when all categories $E,C$ and $C$ are Brown categories of cofibrant objects, the functor are morphisms of categories of cofibrant objects, the pullback is a pseudo-pullback and one of the functor is a "fibration" in the sense of Karol Szumilo work. Because Szumilo shows that Brown categories of cofibrants objects form a category of fibrant objects, pullback along fibration are homotopy pullback and one can deduce the desired result from this. I would expect one can say something about lax pullback from this as well, but I don't think this has been investigated. Also in your setting one would need to assume that $G$ (and $F$) are left or right Quillen functor to hope to apply this kind of idea...

An alternative version of Szumilo's construction can also be find in my own work, where the notion of "fibration" is a little more general. Though with this version it has not been clearly proved that there is an equivalence with an appropriate class of $(\infty,1)$-categories, so it wouldn't be possible to deduce form the homotopy pullback something about a pullback of $\infty$-categories. Of course, this is also true, it would just require some extra work.

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.

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.