For Instrumental Variables, Why can we show that it is unbiased by taking E(Y|X,Z)? My understanding is that instrumentals variables regressions estimator is consistent, but not unbiased, for identifying the causal effect of a variable x and y.
I understand that for an instrument,z, cov(z,error) = 0 is the identifying assumption needed to ensure that the instrumental variable regression estimator is a consistent estimator of the causal effect. But why can't we use E(error|Z) = 0 to show it is unbiased? Further, what is mathematically incorrect about taking the expectation of the error given Z and X? For example, if have the IV estimator as follows:
$\beta_{IV}$ = $\beta$ + $\frac{Cov(Z,error)}{Cov(Z,X)}$
I understand that we cannot show this is unbiased because even assuming E(error|Z) = 0,  E(B|Z) does not allow use to pull the covariance of z and x outside of the expectation, but why can't I condition on Z and X? For Example, E(B|Z,X) = $\beta$

Because $E[\text{error}|Z,X]\neq 0$ in general.
We know that (in general) $E[\text{error}|X]\neq 0$, because that's why we're going to the effort of using instrumental variables.  Since
$E[\text{error}|X]=E_Z\left[E[\text{error}|X,Z]\right]$
it's not possible for $E[\text{error}|X,Z]$ to always be zero.
