Does $Cov(X,Y)=0$ imply that the sample covariance between realizations of $X$ and $Y$ is always zero?
Certainly not! In fact, if you were to generate only two realisations of the pair of variables then you would almost certainly get perfect sample correlation!
To see this, consider two data points $(x_1, y_1)$ and $(x_2, y_2)$ with $x_1 \neq x_2$ and $y_1 \neq y_2$. It can easily be shown that the sample correlation of this data has a magnitude of one! That is, they have a perfect linear relationship! (This happens because any two points on a plane can be connected by a perfect straight line.)
For instance, in linear regression, we have that $Cov(e,\hat Y) = 0$. That is, the residuals and fitted values are uncorrelated. Is this always true in any sample realization of residuals and fitted values?
This is actuall a completely different question to the title question. In the specific case of a regression model using OLS estimation, the residual vector and the fitted response vector will always be orthogonal. This is a stronger property than saying that they have zero covariance in a distributional sense. It occurs because OLS estimation is equivalent to an orthogonal projection onto the column space of the design matrix, which decomposes the response vector into a fitted vector and an orthogonal residual vector.