Formal test to justify the use of sampling weights in survey data Is there a formal method/test/proof that one can use to justify the use of sampling weights when using survey data? You could compare the coefficients in both regressions, the R2, etc. But is this enough?
I ask this because there tests and estimation methods available for my software (Stata) is restricted when using weights. 
Notice that the use of weights many times means that the sample sizes differ among regressions (as some observations have weight equal to zero).
 A: Bollen et. al. (2016) wrote a pretty accessible paper following the suggestions from Fuller (2009) and Pfeffermann (1993) to utilize Hausman (1978) test.
Generally, Hausman test compares two sets of coefficient estimates that are expected to perform differently depending on whether a statistical model is correctly specified. If it is, there usually exists an asymptotically efficient estimate (e.g., a maximum likelihood one). If the model is misspecified, this great estimate often breaks down and may become inconsistent. That's where the other estimates comes to rescue that is robust to model misspecification, and consistent regardless of whether the model is specified correctly or not.
In the current setting, the efficient model is the one without weights (because adding weights usually leads to losses of efficiency known as design effects), while the robust model is the model with the full svy specification. If we are talking about regression, then the simple model is that of homoskedastic regression; the violations include the informative design features, where any of stratification, clustering, and unequal weights may be related to the outcome. Then with some tweaks you can form the Hausman test between the two; in Stata, you may have to use hausman, force option to override Stata's (generally justifiable) thinking that sandwich estimators should not be used in Hausman tests.
P.S. Zero weights sound iffy to me.
