OLS vs IV estimates - Sign and Significance Assume I have an equation with 1 endogenous variable, and many other exogenous variables. Also assume I have 2 valid instruments for the endogenous variable for IV estimation.
If I were to estimate this equation by OLS estimation and IV estimation , would I expect the sign of the EXOGENOUS variables and statistical significance to remain the same?
For a particular exogenous variable, if the signs and statistical significance under OLS and IV are the same, then why is this the case?
 A: My answer is caveated by the questions "Remain the same" as what exactly? and What precisely do you mean by "expect?" 
Assuming you mean if you compare the results of an (uninstrumented, single stage) and an instrumental variables regression of the same outcome, and by "expect" you are asking about the possibility of sign and significance changes, then: 
No. It is very possible that both sign and $p$-value changes. The possibility of changing sign underscores why directed acyclic graphs and the analytic methods accompanying counterfactual interpretations of causality are worth thinking hard about, and also underscores the usefulness of instrumental variables regression in helping to provide unbiased causal estimates if their assumptions are met. If I understand your question correctly, then the below signed DAG represents one possible scenario, where $O$ is the dependent variable, $X$ is an endogenous variable, $Z_{1}$ and $Z_{2}$ are exogenous variables, $U$ is unobserved, time flows from left to right, the arrows with pointed heads mean "causes an increase in (the probability of)", and the arrow with a round head means "causes a decrease in (the probability of)." Here both $Z_{2}$ and $U$ confound $E(O|X)$ as an unbiased estimator of the true causal effect of $X$ on $O$. Adjusting the regression by $Z_{2}$ removes the bias from $Z_{2}$, but that from $U$ remains. Given an exogenous predictor like $Z_{1}$, we could obtain an unbiased estimate of the true causal effect of $X$ on $O$ using $\hat{X}$ (as predicted by $Z_{1}$). 
Sign change in the endogenous case:Because the effect of $U$ on $O$ is negative, $U$ specifically biases $E(O|X)$ downward; whether the bias takes $E(O|X)$ only towards 0, or past 0 and into negatives (sign-switch) depends on the relative magnitude on the effects of $X$ on $O$, $U$ on $X$ and $U$ on $O$.
Sign change in the exogenous case:Because $X$ is a causal "collider" for $Z_{1}$ and $Z_{2}$, instrumenting $X$ only on $Z_{1}$ produces biased estimates both for the effect of $X$ on $O$, and for the effect of $Z_{2}$ on $O$:
For OLS ($O \sim 1 + Z_{1} + Z_{2} + X$):


*

*A larger causal effect of $Z_{1}$ on $X$ relative to the causal effect of $Z_{2}$ on $X$, biases $\beta_{Z_{1}}$ to increase, and biases $\beta_{Z_{2}}$ toward 0.

*A smaller causal effect of $Z_{1}$ on $X$ relative to the causal effect of $Z_{2}$ on $X$, biases $\beta_{Z_{1}}$ downward (possibly into negatives/switching signs) and biases $\beta_{Z_{2}}$ downward (possibly into negatives/switching signs).
For IVR ($O\sim 1 + \hat{X} + Z_{2}$; $\hat{X} \sim 1 + Z_{1}$):


*

*A larger causal effect of $Z_{1}$ on $X$ relative to the causal effect of $Z_{2}$ on $X$, $\beta_{Z_{2}}$ is biased to estimate above its true direct causal effect; $\beta_{\hat{X}}$ is a nearly unbiased causal estimate of the effect of $X$ on $O$ (slightly towards zero).

*A smaller causal effect of $Z_{1}$ on $X$ relative to the causal effect of $Z_{2}$ on $X$, $\beta_{Z_{2}}$ is biased to estimate above its the true (direct) causal effect; $\beta_{\hat{X}}$ is biased towards zero.
For IVR ($O\sim 1 + \hat{X} + Z_{2}$; $\hat{X} \sim 1 + Z_{1} + Z_{2}$):


*

*A larger causal effect of $Z_{1}$ on $X$ relative to the causal effect of $Z_{2}$ on $X$, $\beta_{Z_{2}}$ is a nearly unbiased estimate the true direct causal effect of $Z_{2}$ on $O$; $\beta_{\hat{X}}$ is a nearly unbiased estimate of the true causal effect of $X$ on $O$.

*A smaller causal effect of $Z_{1}$ on $X$ relative to the causal effect of $Z_{2}$ on $X$, $\beta_{Z_{2}}$ is biased to estimate below its the true (direct) causal effect (possibly into negatives/switching signs); $\beta_{\hat{X}}$ is slightly biased to overestimate estimate of the true causal effect of $X$ on $O$.
In the first case for all three models, where the true causal effect of $Z_{1}$ on $X$ is larger than that of $Z_{2}$ on $X$, any biases of $\beta_{Z_{2}}$ do not switch the sign, and the magnitude of the estimate can be different or the same for each model depending on the difference in magnitudes of the true effects of $Z_{1}$ and $Z_{2}$ on $X$.
In the second case for all three models, where the true causal effect of $Z_{1}$ on $X$ is smaller than that of $Z_{2}$ on $X$, the biases of $\beta_{Z_{2}}$ are all capable of switching the sign but the downward bias is different for each model such that the term may be negative in OLS, but positive in one or both of the IVR models.

Even if the sign does not change, one would expect the $p$-value of the endogenous variable to change more often than not, as two-stage IV regression necessarily sacrifices variance and therefore statistical power in the instrumented variable (so in the provided example, specifically $s^{2}_{\hat{X}} < s^{2}_{X}$, because we dispense with the portion of the variance of $X$ that is not explained by $Z_{1}$). One could see the $p$-value of the exogenous effects (e.g., $\beta_{Z_{2}}$) to change between models because the effect itself changes, or not change because the effects are similar. 
Of course the signs do not necessarily have to change (again, it will depend on the relative magnitudes), and the loss of variance in an instrumented predictor ($p$-values increase) may be offset by better model fit due to decrease in bias from that predictor ($p$-values decrease).
