Which supervised machine learning algorithms assume normally distributed feature variables?
If you are distinguishing between Statistics and Machine Learning, then you need to define boundary between the two, and that boundary is going to be opinion-based. It is a matter of definition which algorithms fall into Machine Learning and which under Statistics.
I'll try to give some examples without committing to one or the other.

Linear regression expects the errors (residuals) to be normally distributed. This comes from the maximum likelihood and the $x^2$ term in the normal distribution's PDF.
Logistic regression expects the log-odds of class membership to be linear. This is given for two normally distributed classes with equal variance. It follows from the Bayesian probability.
Linear discriminant analysis expects two normal-multivariate distributed classes with the same covariance matrix.

You can apply these algorithms even if your data violate the assumption, but then the results (in statistics: parameter estimates) won't be (quite) correct. How much they'll deviate from the real values depends on the extent of the assumption violation.