What control variables do we need for IV regression? I am running and IV regression and I need help on what control variables to include in the IV regression. I believe that I have to control for potential variable that are correlated with both the instrumental variable (Z) and the dependent variable (Y) so that exclusion restriction holds. 
What else do I have to control for? In IV papers that I read, I see authors also use control variables that were use in the OLS regression (to prevent omitted variable bias = those correlated to both independent X and dependent variable Y) in the IV regression. If IV regression can solve omitted variable bias, why do we need to include those same variables from the OLS regression in the IV regression? Would we not get the same estimate from IV regression, if we do not include those control variables that were used in OLS?
 A: If IV regression can solve omitted variable bias, why do we need to include those same variables from the OLS regression in the IV regression?
If you can argue that your IV is as good as random then you don't need to include any other variables to remove the bias. In many cases (e.g. randomized controlled trials) this is satisfied by design. But when using instruments in observational studies, it is much harder to show that a instrument is plausibly exogenous. In such cases we relax the assumption and require the instrument to be random after conditioning on some observed variables. In such cases, the IV can remove the omitted variable bias only after the variations due to the observables have been controlled for. So you need to control for this observable characteristics.
Would we not get the same estimate from IV regression, if we do not include those control variables that were used in OLS?
Not necessarily. Firstly, if the IV is conditionally exogenous then not including the controls will bias your results. But what if the IV is perfectly random? Then you get unbiased estimates, but the precision of your estimate is determined by how much of the variation in the endogenous variable is explained by your instrument. Imagine your endogenous variable to have "good" variations and "bad" variations (these are the omitted variable correlations). The bad variation is what you want to get rid off, but you want to capture as much of the good variation as you can. Adding relevant variables will explain a larger part of the good variation and make your estimates more precise. 
Another reason for getting different estimates when using control variables is the idea of Local Average Treatment Effect. Basically using IV allows you study the effect of only the part of the endogenous variable that the instrument can explain. Hence, adding more variables (which are assumed to be exogenous) will give you an effect closer to the average effect of the endogenous variable (please see mostly harmless econometrics).
What else do I have to control for?
I think this is a more philosophical question and depends on the scope of your research. If you have a exogenous instrument $I$ and your motive is to study the causal impacts of an endogenous $D$ on $Y$, then simply using these three will give you the local average treatment effect of $D$.  
