It is true that in the category of spaces there exists a characterization of homotopy pullbacks in terms of homotopy fibers (Proposition 4.1).
I want to know a category (or $\infty$-category) where I can find a square diagram where there is an equivalence in all the homotopy fibers but the diagram is not an homotopy pullback.
For the record, the statement in spaces is this. A diagram $$ \array{ A &\stackrel{f}{\longrightarrow}& B \\ \downarrow && \downarrow^{p} \\ C &\stackrel{g}{\longrightarrow}& D } $$ is a homotopy pullback if and only if, for every point $b: \ast \to B$, the map of homotopy fibers $hofib_b(f) \to hofib_{p(b)}(g)$ is an equivalence.
We could try a direct translation of this into a general $\infty$-category. Suppose that we have an $\infty$-category that has (homotopy) pullbacks and a (homotopy) final object $\ast$. Then, given a homotopy pullback diagram as above, for any map $b: \ast \to B$, the map of homotopy fibers $A \times_B \ast \to C \times_{D} \ast$ is an equivalence. Does this have a converse?
Here is an example where this fails: the opposite category of the category of spaces. In the opposite category, homotopy pullbacks translate into homotopy pushouts and the final object translates into the initial object $\emptyset$. In these terms, the question translates into the following: is a square diagram as above a pushout diagram if and only if, for every map $\epsilon: C \to \emptyset$, the map $B \coprod_A \emptyset \to D \coprod_C \emptyset$ is an equivalence? This criterion does not work, because there are no maps $C \to \emptyset$ unless $C$ itself is empty -- every diagram where $C$ is nonempty satisfies this criterion.
(Another example that works is the category of pointed spaces, because the map $\ast \to B$ is forced.)
I believe that the real key behind the original criterion is that it is not about pullback diagrams. It is about the fact that $\ast$ generates the ($\infty$-)category of spaces: all spaces are built from $\ast$ under homotopy colimits. The same is not true of $\emptyset$ in the opposite category, or $\ast$ in the category of pointed spaces; it is true of $S^0$ in the category of pointed spaces. You can detect whether a square diagram is a pullback by testing it against maps from a collection of generators.
