Let's assume we have three random variables X, Z and Y. If we know that $Cov(X, Z) \neq 0$ and $Cov(X, Y) \neq 0$, without any other restrictions, is it true to say that also $Cov(Z, Y)$ is $\neq 0$.
My gut feeling tells me yes, because I can't think of an example where it wouldn't be like that
Your feeling tells you wrong. For example, let $Y$ and $Z$ be independent non-degenerated and square-integrable random variables. Then $Cov(Y,Z) = 0$ by independence. Let $X=Y+Z$. Then \begin{align} Cov(X,Y) &= E[(Y+Z)Y] - E[Y+Z]E[Y] = E[Y]^2+E[ZY] - (E[Y])^2- E[Z]E[Y]\\ &= E[Y^2]-(E[Y])^2 \neq 0 \end{align} (by independence and since $Y$ is non-degenerated). Similar, $Cov(X,Z) = E[Z^2]-(E[Z])^2\neq 0$.