Suppose we want to estimate $$y = \alpha_0 + x_1 \beta + c'\gamma + \varepsilon$$ where $\beta$ is the variable of interest and $c$ is a vector of controls. We suspect $E[x\varepsilon]\neq0$ but we have an instrument $z$ such that $E[z\varepsilon]=0.$ My question is - as long as IV is valid - should we care that there could be an endogenous control $c_1$ such that $E[c_1\varepsilon]\neq0$? Why or why not?

My hunch is that it doesn't because as long as IV is valid (it satisfies inclusion-exclusion principle) the correlation between controls and unobservables should not lead to the inconsistency of the estimate of $\beta$.

In general, if $c_1$ is endogenous, you need instruments for $c_1$ as well.

Example: Even when $x_1$ is exogenous, if $c_1$ is endogenous and $x_1$ and $c_1$ are correlated, then OLS (which is the IV estimator using $x_1$ as IV for $x_1$) is generally inconsistent.

Exceptions exist. For example, when exactly identified, if $(Ezc') (Ecc')^{-1} (Ecu)=0$, then your IV estimator is consistent, I guess. (Can derive this using standard technique involving the law of large numbers.)