Let's take 3 random variables $X,Y,Z$ on the same probability space (or associated with the same experiment), then applying the tower law in an iterative fashion we get:
$$ E[X] = E_Y[E_{X|Y}[X|Y]] = E_Y\left[E_{Z|Y }\left[E_{X|Y,Z}\left[X|Y,Z\right]|Y\right]\right] $$ where I specified the pdfs on which we integrate below the expectation $(E)$ symbol. Precisely, we have that $$ E_{X|Y}[X|Y] = E_{Z|Y}\left[E_{X|Y,Z}\left[X|Y,Z\right]|Y\right] $$
However, if I am not mistaken, the above implies that the following is not true:
$$ E[E[X|Y]|Z] = E[E[X|Y,Z]] $$
I have some difficulty in understanding why, any help or suggestion would be much appreciated.
No, that is not true in general, because the left-hand side is a random variable and the right-hand side is a number. Suppose $X=Y=Z$ then, $$E(E(X|Y)|Z) = E(E(X|X)|X) = E(X|X) = X$$ but, $$E(E(X|Y,Z)) = E(E(X|X)) = E(X)$$ which are not equal so long as $X$ is not degenerate.