Let $X$ and $Y$ be two random variables and let $z$ be a constant. Is the following true:
$E\Bigr[E[X|Y]\Bigl| Y>z] = E[X| Y>z]$
I think it is true by the Law of Iterated Expectations. Is this correct? If not, is the statement true due to some other property or is it false in general?
It is true
Let $X$ and $Y$ be defined on $\Omega$ and let $(\Omega,\mathcal F, P)$ be our probability space. If $\mathcal D\subseteq \mathcal E$ are sub $\sigma$-algebras of $\mathcal F$ then $$ E[X\mid \mathcal D] = E\bigl[E[X\mid \mathcal E]\mid \mathcal D\bigr] = E\bigl[E[X\mid\mathcal D]\mid \mathcal E\bigr], $$ which we call the law of the iterated expectation.
Note that $\{Y>z\} \subseteq \{Y\in \mathbb R\}$ and recall that what we mean by $E[X\mid Y]$ is the expectation of $X$ given the $\sigma$-algebra of $Y$, so you are fine.