The statement is false in its current form: there are left Quillen functors between stable model categories that do not preserve finite limits.
However, since ∞-categories are mentioned, presumably what is being meant is that the derived functor of a Quillen functor preserves finite homotopy (co)limits.
Without loss of generality we can treat the case of left Quillen functors.
The left derived functor of a left Quillen functor preserves all (small) homotopy colimits.
Thus, it remains to show that the left derived functor of a left Quillen functor between stable model categories preserves finite homotopy limits.
The latter property in its turn boils down to the preservation of the zero object and homotopy fibers.
The zero object is also the initial object, which is preserved by any left Quillen functor.
The homotopy fiber of $f\colon X→Y$ can be computed as the (derived) loop object of the homotopy cofiber of $f$.
The left derived functor of a left Quillen functor preserves homotopy cofibers, so it remains to show that it commutes with the derived loop object functor.
By definition of a stable model model category, the derived loop object functor is the inverse of the derived suspension functor, and the latter functor is computed using a homotopy pushout, which is preserved by the left derived functor of a left Quillen functor. This completes the proof.