Suppose we have the long regression
$$ y = \alpha + \beta D + \gamma X + \varepsilon $$
but instead use the short regression
$$ y = \alpha + \beta D + \varepsilon $$
then one can show that the OLS estimator is inconsistent and the bias depends on the covariance between $X$ and $Y$ and the covariance between $X$ and $D$
$$ \hat{\beta}_{OLS} \overset{p}{\to} \beta + \gamma \delta $$
where $\delta$ is the coefficient from a regression of $X$ on $D$. This is covered in many Econometrics classes.
What about the case when the long regression also includes an interaction effect (presumably capturing selection on gains)
$$ y = \alpha + \beta D + \gamma X + \eta D \times X + \varepsilon $$
and we omit both terms from the short regression?