Does $Cov(X,Y)=0$ imply that the sample covariance between realizations of $X$ and $Y$ is always zero? 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? I think the answer is yes, but I'm having some difficulty intuitively seeing why the population property carries over to every sample.
Thanks for any help.
 A: 
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.
A: The question in the title is different from the example provided and has a different answer.

If $\operatorname{cov}(X,Y)=0$, is the sample covariance between samples from $X$ and $Y$ also necessarily zero?

The answer to this is No, the sample covariance need not be zero. The sample covariance is a random variable with mean zero and should be expected to have small value (at least compared to $\sigma_X\sigma_Y$), but it is not necessarily exactly equal to zero.

In a linear regression, is the vector $e$ of residuals orthogonal to the vector of estimates?

The answer to this question is Yes.  See this answer (or the Wikipedia entry cited in a comment by @Ben on his own answer) for a geometric viewpoint as to why this is so.
