If we look at the proof that I know of in the case this is true - see below - we need in fact that, for any small $1$-category $I$, $h_\infty(C^I)$ has small (co)limits and that they can be computed termwise, and we need to prove the particular case where $I$ is discrete as a first step, which usually comes from a variant of calculus of fractions ($C$ could be an $\infty$-category to begin with). The proof can be formulated in a model free way once we have a good theory of pointwise Kan extensions.
Anyway, this is true for any model category (in Quillen's original sense) $C$ which has either small colimits or small limits. For a category with fibrations and weak equivalences $C$ (such as a category of fibrant objects à la Brown), and if small products of (trivial) fibrations between fibrant objects exist and are (trivial) fibrations (this already implies that $h_\infty(C)$ has small limits), this is true if one of the following further conditions is satisfied:
- limits of countable towers of (trivial) fibrations between fibrant objects are (trivial) fibrations;
- the factorization of any map with fibrant codomain into a weak equivalence followed by a fibration can be made functorially.
In fact the last two conditions are only there to ensure that $C^I$ is a category with weak equivalences and fibrations defined level-wise (the tricky part being the existence of factorizations if ever these were not functorial in $C$ to begin with, but this is documented in the work of Radulescu-Banu). The principle of all the proofs I know of is that one proves that both $\infty$-categories $h_\infty(C^I)$ and $h_\infty(C)^I$ have small limits (it sufficices to prove this is true for $h_\infty(C)$ for a generic $C$), and to prove that the comparison functor $h_\infty(C^I)\to h_\infty(C)^I$ commutes with small limits (=with small products and with pullbacks). One can then check fully faithfulness on objects which are homotopy right Kan extensions of objects of $C$ along an object of $I$ (seen as a functor from the terminal category), and prove through cofinality arguments that any $I$-indexed functor is an homotopy limit of such thing to finish the proof. This kind of approach survives very well if we allow $C$ to be an $\infty$-category to begin with (but $I$ must be a $1$-category), so that any reasonnable notion of model $\infty$-category (e.g. Mazel-Gee's) is eligible for such rectification theorem. The case where $I$ is a finite direct category holds with a much greater generality (no need to have small homotopy (co)limits), but the idea of the proof is similar (with simpler technical arguments, though). References can be found in my book on higher categories: Theorem 7.9.8 (with Example 7.9.6 and Remark 7.9.7. to get easy sufficient conditions in practice) if $I$ is mall and Theorem 7.6.17 if $I$ is a direct finite category. It is striking to me that this kind of rectification is the best explanation of why homotopy limits in $C$ coincide with limits in $h_\infty(C)$; see Remark 7.9.10. In the same direction, there is Balzin's work which is more general because it deals with sections of general fibred model structures but also less general because it restricts to the case where the indexing small category is Reedy.