Does Naive Bayes Classifier require assuming a specific distribution for training?
The most harrowing problem would be "interpolation". That is, computing $p(x_i | y)$ for $x_i$ and $y$ you have not yet observed, but fall within the convex hull of your observations. 

You have to bin events together, or come up with a wacky interpolation scheme. This does work. However, If you assume a distribution, $p(x_i | y)$ becomes continuous, and you end up with a more statisically sound means of interpolation.