Multiple Linear Regression Zero Conditional Mean Assumption Greene [1] and Wooldridge [2] emphasize that in the standard multiple linear regression model 
$${\bf y}=X{\bf b}+{\bf e}$$
a key assumption is that
$$E[{\bf e}|X]=E[{\bf e}].$$
Or, in other words, $X$ provide no information about the expected value of ${\bf e}$. Provided that we include an intercept in the model, this assumption will be equivalent to 
$$E[{\bf e}|X]=E[{\bf e}]={\bf 0}.$$
Wooldridge call it zero conditional mean assumption.
On the other hand, Gelman and Hill [3 (p. 46 kindle edition)] list the assumptions of linear regression in order of importance as


*

*Validity

*Additivity and linearity

*Independence of errors

*Equal variance of errors (for the efficiency of the estimator)

*Normality of errors (for small sample inference of variance of the estimator).


Note that, at this point in the book Gelmen and Hill have not discussed the use of linear regression as a tool for causal inference yet. In the discussion of causal inference, in chapter 9, they introduce ignorability assumption that
$$y_0, y_1 ⊥ T | X.$$
The ignorability assumption seems to be closely related to zero conditional mean assumption.
The Questions


*

*Why Gelman and Hill did not include zero conditional mean at the begining?

*How can we interprete OLS estimator of ${\bf b}$ when zero conditional mean is violated?


References
[1] Greene, William, 2008, Econometric Analysis, 6th Edition, Pearson.
[2] Wooldridge, Jeffery, 2015, Introductory Econometrics, 6th Edition, Cengage Learning.
[3] Gelman, Andrew, and Jennifer Hill, 2007, Data Analysis Using Regression and Multilevel/Hierarchical Models, Cambridge University Press.
 A: I'll start with your second question as it will inform the answer to the first.

Note the distinction between regression coefficients and structural causal model coefficients. The former is what you get when you run a regression - always. Only under specific circumstances would the regression coefficients have a causal interpretation, or in other words, only under specific circumstances will the regression coefficients coincide with the coefficients in the structural causal model. What are these specific circumstances? A necessary condition is the zero conditional mean assumption (pertaining to the structural errors), discussed by Wooldridge and Greene. Or the ignorability assumption discussed by Gelman and Hill. The latter is once again a necessary assumption for the regression coefficients to have a causal interpretation, it is just described in a different context - that of potential outcomes. The zero conditional mean assumption and the ignorability assumption, also called selection on observables, and also called CIA [Conditional Independence Assumption] (in Mostly Harmless Econometrics) are two sides of the same coin. Chen & Pearl said with reference to Greene's book "In summary, while Greene provides the most detailed
account of potential outcomes and counterfactuals of all the authors surveyed, his failure to acknowledge the oneness of the potential outcomes and structural equation frameworks is
likely to cause more confusion than clarity, especially in view of the current debate between two antagonistic and narrowly focused schools of econometric research (See Pearl 2009, p. 379-380)."
So, to answer your question. If the zero conditional mean assumption (with regards to the structural errors) is violated then the regression coefficients will not coincide with those of the structural model; in other words, the regression coefficients will not have a causal interpretation.

Because they chose to describe the conditions necessary for the coefficients to have a causal interpretation in the context of potential outcomes. Just the other side of the same coin.
For more detail on the difference between regression and structural causal model, see Carlos Cinelli's answer here and here.
A: I subscribe the answer of ColorStatistics but let me add something. I have faced several problems like your exposed above and actually the key that bring us to solve them is to distinguish clearly between structural (causal) and regression (statistical) quantities. Unfortunately many econometrics books are unclear about that. This unclearness produce ambiguities and sometimes contradictions and mistakes. What you write/report is an example:

Greene [1] and Wooldridge [2] emphasize that in the standard multiple
linear regression model  $${\bf y}=X{\bf b}+{\bf e}$$ a key assumption
is that $$E[{\bf e}|X]=E[{\bf e}].$$ Or, in other words, $X$ provide
no information about the expected value of ${\bf e}$. Provided that we
include an intercept in the model, this assumption will be equivalent
to  $$E[{\bf e}|X]=E[{\bf e}]={\bf 0}.$$ Wooldridge call it zero
conditional mean assumption.

All the above is referred on linear regression model. The fact that the insert of the constant is considered as relevant assumpion underscore even more clearly that all above are written on regression equation. Those core  assumptions are given at the start of the books and have consequence on all chapters ahead. No causal concepts/assumptions are given, at least not clearly, therefore no clear causal conclusions can be found. Unfortunately causal concepts are, or should be, one pillar in econometrics and in fact these, or something like these, appear in the books.
The zero conditional mean assumption for the error term, usually called exogeneity (even in Greene and Wooldridge), should be referred on a structural error, therefore a causal model should be involved.
Matter of fact is that in regression equation the zero mean for error and orthogonality between regressors and error holds by costruction and not by assumption; note that these fact remain true even at population level. Zero mean conditional independence between error and regressors can fail in regression equation. However if the exact conditional expectation form is linear the usual condition ($E[\epsilon|X]=0$) hold by construction too. In the case of regression on gaussian data even stochastic independence, between regressors and error, hold. However we have to note that not even this strong statistical condition is enough for achieve causal conclusions (see here:non stochastic regressors)
Even detail about contemporary observations or cross-observations can appear (see strict vs weak exogeneity). I faced all the possibility years ago for the first time. All of these thing tend to complicate the problems and obscure the key point that if we want to face the causal problems we need a causal model. Detail about statistical conditions cannot be enough, never. I surveyed many econometrics books in several edition. Is not easy to find authors/books the share precisely the same assumptions and terminology, several differences appears. However it seems me that there are no one  econometrics book yet that use all the tools developed in causal inference literature, in most case no one is properly used. Read here: How would econometricians answer the objections and recommendations raised by Chen and Pearl (2013)?
I wrote something about this and related topic in this site. Among others:
linear causal model
Structural models and relationship (statistical associations)
Random Sampling: Weak and Strong Exogenity
Regression and causality in econometrics
What is the actual definition of endogeneity?
Zero conditional expectation of error in OLS regression
maybe these can help.
All I said can be useful for understand that is not easy to give a proper role to the zero conditional mean assumption as written above. In structural equations (causal models) it is crucial as causal assumption. At the other side, even if it can sound strange, in pure regression this assumption imply  only restrictions on the joint distribution of the data; usually not a core question in econometrics. About the assumpions written as in the book of Gelman and Hill, note that in pure regression linearity assumption imply $E[\epsilon|X]=0$ also.
