Here is the example I always give to the students. Take a random variable $X$ with $E[X]=0$ and $E[X^3]=0$, e.g. normal random variable with zero mean. Take $Y=X^2$. It is clear that $X$ and $Y$ are related, but
$$Cov(X,Y)=E[XY]-E[X]\cdot E[Y]=E[X^3]=0.$$
