In the case of an $\mathbb N^{op}$-indexed system specifically, the answer is yes (note that this is implicit in the Stacks project link you gave); in fact if you replace "fiber sequence" by "equalizer", this holds in an arbitrary $\infty$-category with the appropriate limits (namely products and equalizers). The description in terms of fibers does not hold in general though (in an arbitrary $\infty$-category with limits, the "fiber" without specifying a basepoint does not even make sense)

There are several ways a proof could go. One approach is to use the fact that $\mathbb N$, as an $\infty$-category, is the infinite pushout $[1]\coprod_{[0]}[1]\coprod_{[0]}[1] \dots$. Another approach is to use the Yoneda lemma to reduce to the case of the $\infty$-category of spaces, and there, use explicit models for homotopy limits (this is probably simpler to actually write down, if not as conceptual). 

For a general filtered poset $I$, however, homotopy limits over $I^{op}$ can be more complicated. 

There is always the general "Bousfield-Kan formula", which expresses $\lim_{I^{op}}$ in the form of the totalization of a cosimplicial object (the $\infty$-analogue of an equalizer): namely, $\lim_{I^{op}}F$ can be described as the limit over $\Delta$ of a functor that looks like $[n]\mapsto \prod_{i_0\to ... \to i_n} F(i_0)$ .