Assume, we have the following:
$Y$- outcome
$D$- exposure
$U$- unobserved confounder
$V$- instrumental variable.
Assume there are no observed confounders.
Can we check the assumption of exclusion restriction by running a regression of $Y$ on $D$ and $V$ and testing if the coefficient associated with $V$ is 0? The rational is all the effects of $V$ on $Y$ has to go through $D$ so once $D$ is controlled, there should be no effect of $V$.
Can we check the assumption of exclusion restriction by running a regression of Y on D and V and testing if the coefficient associated with V is 0?
No, you can't. It's easy to see why visually. Consider the DAG below, representing your set up:
Your proposal is to regress $Y$ on $D$ and $V$ and check if the coefficient of $V$ is zero. But look what happens if you condition $D$:
That is, even though you have blocked the directed path $V \rightarrow D \rightarrow Y$, you now opened the path $V \rightarrow D \leftarrow U \rightarrow Y$ (represented in red)! The treatment, in this case, acts as a collider, and conditioning on it opens the mentioned spurious path.
Thus, we should not expect the regression coefficient of $Y$ on $V$ conditional on $D$ to be zero even if $V$ is a valid instrument.