Why is it necessary to evaluate stationarity and seasonality of model residuals? Or is it? The model in question is an OLS model that represents a relationship between Y and a bunch of economic variables.
OLS is only supposed to work (and deliver minimum variance unbiased estimates) under a set of assumptions. If you run OLS but do not check the assumptions, you will not know whether you can trust the results. Therefore you want to check the assumptions.
Model residuals being non-stationary (for example, due to having a seasonal component in them) is a violation of the assumptions. Some forms of non-stationarity may be less harmful than other; if residuals are only heteroskedastic, OLS will still deliver consistent estimates (ones that converge to the true values when the sample size grows) although they will no longer be minimum variance (a.k.a. efficient); if residuals follow an integrated process (a.k.a. unit-root process), OLS is no longer consistent (am I right on this one?), and that is something you really do not want. Generally, if you see some pattern (like seasonality) in the residuals, it indicates that you have misspecified the model. The cure would be to think about model specification again (e.g. model the seasonality explicitly).