Why do error values in linear regression have to be normally distributed and why not in logistic regression?
Lets clarify with a simpler question
Why points on a circle must be equally distanced from center, but not points on a square?
Because of how circle and square are "defined". The same goes for linear and logistic regression, we cannot pose a "why?" question when errors are "defined" in a certain way (or a better word here "assumed"). We may design a new version of linear regression by replacing Normal distribution with some other distribution, and then proceed to derive a formula or algorithm for estimating the parameters. Then use it in a real-world scenario to see how it works, and so on.
Also here is a list of good posts on stats.stackexchange.com related to characteristics of those errors: