What will happen if I use a nonparametric test with normally distributed data? I was asked if a type I error in the Shapiro-Wilk test would impact the main analysis and if the wrong test was used if it would matter or not if my data was normally distributed... 
 A: In statistical analysis, if your data follow a parametric distribution, you should utilize the benefit of knowing the distribution, and employ the statistical methods based on that distribution. 
But sometimes we do not know the distribution of the random variable, so the nonparametric statistical methods were developed to embrace the wide range of the distributions while sacrificing some efficiency.
Given you know the distribution of random variable and use the nonparametric statistical method, instead of parametric statistical methods based on knowing the distribution, it will be inefficient, i.e., the power of test will decrease, standard error will increase, and the confidence intervals will be wider than with the parametric method.
A: If your data happened to be drawn from a normal population (and the other usual assumptions for an ordinary t-test apply), then the test works as it should (it's non-parametric, it's supposed to work). There's no drama on that score. 
If you know enough that you're confident in assuming normality you may want to take advantage of that knowledge, but for many tests it doesn't help you a lot.
If you're doing one of the common location-tests (Wilcoxon signed rank test, Wilcoxon-Mann-Whitney test) you lose almost nothing (power-wise) in a test for a location shift by ignoring the normality. [You need one extra observation for every 21 observations to match the power of the most powerful test when all its assumptions hold.]
If you're dealing with some other tests is may matter a bit more (though some may matter even less). One example where it makes a somewhat bigger difference is using a Friedman test compared to the corresponding ANOVA test in a randomized blocks design.
