This question arose while reading Peskin and Schroeder, specifically, it arose in regards to the sum of diagrams above their Eq. (10.20) on pg. 326.
The context is $\phi ^4$ theory and they are using a vertex renormalization condition to compute the counterterm $\delta _\lambda$ corresponding to the coupling constant $\lambda$. To do this, they calculate the $4$-point amplitude up to one-loop order in perturbation theory. In the process of doing this, however, they do not seem to include any one-loop diagrams involving the counterterms. Indeed, they only seem to include counterterm diagrams up to tree level.
Why is this? It seems, at least naively, that if one is doing a one-loop computation, one should compute all one-loop diagrams, and not discriminate between those Feynman rules in the 'original' theory and those that only arise during renormalization.