Do I need to adjust for confounding when the confounder is not causal? Suppose I have a model like 
$$y =\alpha + x_1\beta $$
and that there exists another variable, $x_2$, that is correlated with both $y$ and $x_1$. However, changing $x_1$ will cause changes in $x_2$ but changing $x_2$ does not do anything to $x_1$. Do I need still need to add $x_2$ to the model if what I'm interested in is the relationship between $x_1$ and $y$?
 A: You first need to define precisely what is the causal effect you want to estimate and to lay down your causal assumptions (a causal model).  For instance, you say that $x_1$ causes changes in $x_2$, that means $x_2$ is potentially a mediator (see the third example in this answer). 
Thus, if you want to measure the total effect of $x_1$, you do not want to adjust for $x_2$ in the regression (although depending on context, like in the front-door criterion, you can use mediators for estimating total effects, but this requires more than a simple regression adjustment). 
On the other hand, if you want to measure the direct effect of $x_1$ (holding $x_2$ fixed), then you might need to adjust for $x_2$ in your regression.  But notice that even if $x_1$ was randomized, if $x_2$ was not it might be confounded, and identification of the direct effect will not be possible. Thus, we need to know what your causal assumptions are.
The problem of which variables to include in regression adjustment for estimating total effects has been solved---you should include those variables that satisfy the backdoor criterion.
