For a single factor experiment we have: $$Y_{ij}=\mu+\epsilon_{ij} = \mu_i +\tau_i +\epsilon_{ij}$$, where $\epsilon_{ij} \sim N(0,\sigma^2)$. To make things simpler lets say our experiment is balanced so $i=1,\cdots,k$ and $j=1,\cdots,n$.\ Now, say we have:
$$Cov(Y_{ij}-\bar{Y}_{i.},Y_{i'j'}-\bar{Y}_{i'.}) = Cov(Y_{ij},Y_{i'j'}) + \cdots + Cov(\bar{Y}_{i.},\bar{Y}_{i'.}).$$ Then I considered the cases $i=i',j=j'$ and so on. I only got for $i=i',j=j'$, the covariance is $\frac{N-1}{N}\sigma^2$ and the rest of it is $0$. I'm not sure if its correct or not.