In combinatorial model categories finite limits commute with (sufficiently large) filtered homotopy colimits. Suppose, for simplicity, that the combinatorial model category is simplicial and generating cofibrations have $\lambda$-presentable domain and codomain. In this case $\lambda$-filtered colimits are homotopy colimits. Suppose, in addition, that the underlying locally presentable model category is $\lambda$-locally presentable. Then $\lambda$-filtered colimits commute with $\lambda$-small limits. Finite limits are $\lambda$-small for all $\lambda$.

Lets say that the category $J$ indexing the homotopy limit is finite if it has finitely many objects and morphisms, and the diagram $EJ$ of simplicial sets serving as a cofibrant replacement of the constant diagrams of points indexed by $J$ in the projective model structure on ${\cal S}^J$ has a finite simplicial set in each entry. For example, a finite group is not a finite category by this definition. A finite homotopy limit is a homotopy limit over a finite diagram.

Suppose that $\cal M$ is a simplicial $\lambda$-combinatorial model category and $F\colon J\to \cal M$ a finite diagram. Then $\mathrm{holim}_J F$ may be computed as a weighted limit with the weight $EJ$, in other words this is an end construction:
$$
\mathrm{holim}_J F = \mathrm{hom}(EJ, F),
$$
which is a finite weighted limit commuting with $\lambda$-filtered colimits, hence, commuting with $\lambda$-filtered homotopy colimits. In particular, if $\lambda=\aleph_0$, then filtered homotopy colimits commute with finite homotopy limits.