So, I was thinking before that this might have some nice, simple topos theoretic explanation, but Jacob disabused me of that notion. However, I'm still very interested in the following question:

Is there a criterion for determining when a morphism $f:A\to B$ of connective $A_\infty$-ring spectra is such that $A$ can be recovered as the (homotopy) limit of the (homotopical) Amitsur complex on $f$ in the $(\infty,1)$-category of $A_\infty$-ring spectra? For what it's worth, in the context of EKMM, such a criterion is given as an (unproven) proposition in a paper of Carlsson's (Proposition 3.3 here). There, it is stated that a morphism $f:A\to B$ of connective associative $\mathbb{S}$-algebras has the desired property if $\pi_0(f)$ is an isomorphism and $\pi_1(f)$ is onto. I'm interested in proving this in the $\infty$-categorical context (and really proving it at all).