So I have got 2 variables, $A$ and $B$. I know for a fact that $E(A)$ = 0.
I don't know if they are independent.
$$Cov(A, B) = E(A B) - E(B) E(A)$$
I know that $E(A) = 0$.
Does that mean $E(A B) = 0$, too?
(I'm doing this question where I have to show the covariance between estimated residuals and each of the regressors must always be zero, ).
No, you still have to compute $E(AB)$. To show that the covariance of regression residuals and regressors is zero, write the sum of the squares and differentiate it with respect to the coefficients. Each derivative must be zero for the regression solution. If you look at the derivative you'll see that it is exactly $E(AB)$ (here $A$ is the residuals, $B$ is the regressor), so it must be zero. Hence since $E(A)=0$ you get that the covariance is zero too.