@YuriSulyma in case this helps in searching the literature, there are these things in the world of classical rings called "milnor squares". see e.g. section 1 of arxiv.org/pdf/1612.00418.pdf

@DylanWilson it does for ordinary commutative rings, e.g. $\mathbb Z_p \to W(k)$ on top and $\mathbb F_p \to k$ on the bottom, with $k$ a perfect $\mathbb F_p$-algebra, or Proposition 3.2 of www-personal.umich.edu/~bhattb/teaching/prismatic-columbia/…. I had briefly convinced myself the first one would work with the top line replaced by spherical Witt vectors, but considering higher homotopy groups immediately shows that is nonsense :(

@YuriSulyma It is true if one of the maps is "Tor-unital" or "idempotent" (mathoverflow.net/questions/298435/idempotent-ring-spectrum/…) or "solid" (mathoverflow.net/questions/214559/…). This is easy but a reference is Tamme's paper on excision in K-theory (Compositio). However this condition is quite rare; a pro-version of this holds much more generally, see his paper with Land in Annals (K-theory of pullbacks).