I have 200 discrete, well-spaced plots with reasonably independent sampled values from which I've derived a regression equation. If I use it to predict values on a similarly sized fishnet grid, how much does it matter that neighboring grid cells are correlated in the predictor and response variables and is there anything to be done about it?
The question is a little vague since I don't know exactly what model you are fitting. But I would guess that neighbouring prediction cells are correlated because they have similar covariate values.
The observations $y_i$ should be independent, given the linear predictor. Of course - this assumption should be checked. Often you can do this with a plot of residuals (depending on the model) - these should show no discernable spatial pattern. If there is a pattern consider adding a spatially structured random effect (see GAMs or Gaussian Random Fields)