In regression model

$y_i=\mathbf{x}_i'\beta+u_i$

the usual assumption is that $(y_i,\mathbf{x}_i,u_i)$, $i=1,...,n$ is a iid sample. Under assumptions that $E\mathbf{x}_iu_i=0$ and $E(\mathbf{x}_i\mathbf{x}_i')$ has a full rank, ordinary least squares estimator

$\widehat{\beta}=(\sum_{i=1}^n\mathbf{x}_i\mathbf{x}_i')^{-1}\sum_{i=1}\mathbf{x}_iy_i$ 

is consistent and assymptoticaly normal. The expected covariance betweeen residual and response variable then is

$Ey_iu_i=E(\mathbf{x}_i'\beta+u_i)u_i=Eu_i^2$

If we furthermore assume that $E(u_i|\mathbf{x}_1,...,\mathbf{x}_n)=0$, we can calculate the expected covariance between $y_i$ and the estimate of regression residual:

\begin{align*}
Ey_i\widehat{u}_i&=Ey_i(y_i-\mathbf{x}_i'\widehat{\beta})
\end{align*}
\begin{align*}
=E(\mathbf{x}_i'\beta+u_i)(u_i-\mathbf{x}_i(\widehat{\beta}-\beta))
\end{align*}
\begin{align*}
 =E(u_i^2)(1-E x_i (\sum_{i=1}^n\mathbf{x}_i\mathbf{x}_i')^{-1}\mathbf{x}_i)
\end{align*}

Sorry for poor formatting in the last formula, Mathjax did not want to recognize the line ending sign, and for some reason did not render formula if I used $\mathbf{x}_i$ instead of $x_i$.