What happens if the "control variables" are also endogenous? I work in Political Economy, and a lot of the models include "innocent" control variables such as population, inequality, colonial legacy, etc. so that the author can claim unbiasedness on their independent variable of interest.

But if any of these control variables are endogenous to some omitted variable, doesn't this contaminate the unbiasedness of ALL the independent variables?

If that's true, then what can we do? Leave those control variables out and they lead to omitted variable bias themselves. Include those in and they will contaminate everything in the model.

Example: A researcher wants to know if inequality leads to violence, and he controls for a few things:
\begin{equation}
Violence = Inequality + Growth + Development + \epsilon
\end{equation}
Seeing that Inequality is likely to be endogenous (because of the omitted variable Level of altruism), he will try to find a instrumental variable for Inequality. But aren't Growth and Development likely to be endogenous (i.e. correlated with Level of altruism) too? 

This example may look silly, but my point is in Political Economy / Development work, there are so many factors at play (yet omitted) that I'm afraid many variables included on the LHS are endogenous. Yet often, the researcher only looks for an instrument for his pet independent variable only.
"But if any of these control variables are endogenous to some omitted variable, doesn't this contaminate the unbiasedness of ALL the independent variables?"

I don't want to emphasize this too much, but it's worth mentioning that this is not true in general. The following derivation will hopefully provide some understanding of the "contamination" you mention. As a simple counterexample, suppose that the data generating process is given by
$$
Y = X_1 \beta_1 + X_2 \beta_2 + Z \gamma + \varepsilon,
$$
where $Z$ is unobserved. Let $Cov(X_1,Z) = 0$, $Cov(X_2, Z) \neq 0$, and 
$Cov(X_1,X_2) = 0$. Then, it
is clear that $X_2$ is "endogenous." But notice that because $Cov(X_1,Z) = 0$, our
estimate of $\beta_1$ will still be ok:
$$
\text{plim}\, \hat \beta_{1} = \beta_1 + \gamma \frac{Cov(X_1^*, Z)}{Var(X_1^*)} = \beta_1,
$$
where $X_1^* = M_2 X_1$ and $M_2 = [I - X_2(X_2'X_2)^{-1}X_2']$. Because $Cov(X_1,X_2) = 0$, $X_1^* = X_1$. So $Cov(X_1^*,Z)=0$. 

"What can we do?"

One of the mains challenges of doing good econometrics is thinking of potential identification strategies. In the type of situation you describe, there is probably nothing you can do but to try to approach the problem a different way.