For example in count data modeling, imagine modeling count data generated from dispersed negative binomial using Poisson distribution. The poisson regression fits the significant covariates and they turn out to be strongly significant. Yet the goodness of fit test fails, because the underlying distribution is wrongly assumed. Without assuming knowledge of the negative binomial underneath, how can I proceed with the Poisson regression, what are the limitations or is it even sound to proceed when the GOF test fails?
It shows that one of your basic assumptions is wrong.
However, in practice it depends on how sensitive the parameter you are analyzing is with respect to the underlying assumptions. If you are looking at a strong effect (say, male wage compared to female wage), then whatever you do, the effect is there and about the correct size. One the other hand, if the effect is weakly identified, your results may end up being extremely sensitive to the distributional (and other) assumptions.
Note also that on big data you can pretty much reject every parametric assumption.