Here is the example I always give to the students. Take a random variable $X$ with $EX=0$ and $EX^3=0$, e.g. normal random variable with zero mean. Take $Y=X^2$. It is clear that $X$ and $Y$ are related, but

$$cov(X,Y)=EXY-EX\cdot EY=EX^3=0.$$

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

$$Cov(X,Y)=E[XY]-E[X]\cdot E[Y]=E[X^3]=0.$$

Here is the example I always give to the students. Take a symmetrical random variable $X$ with zero mean (such as normal). Then $EX=0$ and $EX^3=0$. Take $Y=X^2$. It is clear that $X$ and $Y$ are related, but

$$cov(X,Y)=EXY-EX\cdot EY=EX^3=0.$$

Here is the example I always give to the students. Take a random variable $X$ with $EX=0$ and $EX^3=0$, e.g. normal random variable with zero mean. Take $Y=X^2$. It is clear that $X$ and $Y$ are related, but

$$cov(X,Y)=EXY-EX\cdot EY=EX^3=0.$$