Let me preface by saying that I am a relative novice in these matters. In particular, if something below is confusing, do point out, since it is likely that it is a reflection of me screwing something up.
Further, to avoid any confusion (this tripped me up initially) let me specify that a finite category will mean a category with finitely many objects and finitely many morphisms. A finite simplicial set will mean the (usual): a simplicial set with finitely many non-degenerate simplices (presumably, a finite $\infty$-category is a finite simplicial set, but I will let the experts comment on this).
In particular, the nerve of a finite category need not be a finite simplicial set. For instance, the nerve of a (non-trivial) finite group (viewed as a category with one object) has at least one non-degenerate simplice in every degree.
Now let me proceed in increasing levels of generality.
- First, assume that all simplicial sheaves involved are in fact sheaves of (1-)groupoids. So in particular, all sections are Kan. In this case, homotopy limits over a finite category $I$ may be computed correctly without global fibrant replacement. The reason being that in this case filtered colimits commute with the relevant internal mapping spaces. This can be seen directly/elementarily by noting that said mapping spaces are categories of functors and just sitting down and thinking about what the data of said functors (and natural transformations between them) amounts to (picking a finite number of objects, a finite number of morphisms, satisfying a finite number of relations, in the codomain category).
A (seemingly unrelated) earlier post of mine is very closely related to this:
Homotopy fixed points of involutive automorphisms of discrete groups
You can make a bit more of a general statement in this setting by working with indexing categories that along with all overcategories are finitely presented in a certain sense, but I dont know how to make a really clean/simple/practically useful statement here (essentially comes down to characterizing compact objects in the category of groupoids).
- The claim in 1) is also special case of the following. If all sections of the sheaves appearing in the diagrams are $n$-coskeletal Kan complexes (for some fixed $n<\infty$), then homotopy limits over a finite category $I$ may again be computed correctly without taking a global fibrant replacement. The argument is again that filtered colimits commute with the relevant internal mapping spaces. This is true because using the (internal) adjunction between the $n$-skeleton and $n$-coskeleton functor reduces us to the situation where the domain in the relevant internal mapping spaces of simplicial sets is in fact a finite simplicial set (which is a compact object, etc.). There is some commuting of coskeleton functors past certain filtered colimits that I am brushing under the rug here, but I don't think any problems occur (hah! famous last words).
Assuming I have not made an error so far, an interesting "Corollary" of the above is that in these situations the homotopy limit doesn't really depend on the topology of the site. In fact, you can make this completely precise by working with presheaves and noting that this allows you to get the correct "homotopy type" by just computing the homotopy limit at the presheaf level.
- For arbitrary simplicial sheaves you are out of luck without imposing some restrictions on the site. I.e., the only way to get the right homotopy limit without global fibrant replacement is to assume that the nerve of the indexing category is a finite simplicial set (now the relevant commutation of filtered colimits with mapping spaces is well known and just comes down to the characterization of finite simplicial sets as compact objects).
As to restrictions on the site that would get this to work (again, under the assumption that the indexing category is finite)? I haven't yet had a chance to properly think about this, but it's a follow your nose game: you want certain filtered colimits to stabilize. This seems to essentially come down to a finite cohomological dimension type condition (I think it works ok for the big Zariski and analytic sites, maybe also the etale site - but don't quote me on any of this so to speak).
If the indexing category is not finite (say infinitely many objects), then I doubt anything can be said apart from some very specific situations - you are in trouble even commuting a filtered colimit past the first (infinite) product in any explicit model of the homotopy limit (let alone worrying about the horrors that await when dealing with mapping spaces).
Aside: I have now seen a number of places in the literature where it is claimed (erroneously) that homotopy limits over finite diagrams commute with filtered colimits (with the authors pretty clearly meaning finite diagram = finitely many objects and morphisms). Thankfully, in most cases the only homotopy limits considered are actually over diagrams satisfying the stronger finiteness condition (the diagrams are usually just pullbacks). I am coining it the canonical error
It'll be quite ironic if I have made it above in the special situations mentioned!