Throughout, I'll omit the "$\infty$" from the term "$\infty$-category".
It is well-known (and sometimes even included in the definition, although not by Lurie) that pushouts and pullbacks agree in stable categories. It is easy to show that the same property holds (i.e. that limit and colimit diagrams coincide) for higher-dimensional cubical diagrams (i.e. diagrams indexed by the power poset of a finite set). Less obviously, the same holds for diagrams indexed by the slice category $\mathcal{J}_{/V}$ where $\mathcal{J}$ is the category of finite-dimensional positive definite inner-product spaces with morphisms given by isometric embeddings.
Is it known for what other categories this property holds? Obviously for the question to be sensible, we must mean categories with initial and terminal objects. We should also assume that our indexing categories are finite, since stable categories are only assumed to be finitely complete and complete.
Suppose a category $D^{\vartriangleleft \vartriangleright}$ (where the triangles indicate the free insertion of terminal and initial objects to $D$) has this property. I believe that the nerve $N(D)$ (sometimes also called the classifying space of $D$) of $D$ must have the stable homotopy type of a sphere, but I have not carefully checked this, and even if it is necessary, I doubt that it is sufficient.
