I will answer the question in the Neural Network context (i.e. I won't talk about regularization in regression algorithms handles this problem).
The problem of not dropping one of the encoded variables will cause multicollinearity. That is, one of the variables can be estimated using others. e.g. If you have a variable that says whether a person is female or not, why would you need another variable which says whether a person is male or not. However, the main thing that people forgot to say while answering this question is that multicollinearity is not actually a big problem unless you need the interpretation of your variables. Multicollinearity will cause your coefficients to be false, but it won't affect your predictions. Thus, it should not be a big problem unless you need to explain your model in terms of your variables. Another possibility is that, if the covariance between your variables in the training set and test set would be different, then your predictions would be affected, and you would have incorrect results. However, if you shuffle your dataset good enough (which is your assumption) and split the train and test set (also validation) correctly then covariance should be the same in both. In other words, the relation between those correlated variables can be assumed to stay the same in the train and test set. Thus, you can safely assume that your predictions are correct.