Does training a model to minimize the MSE assume that the model predictions are normally distributed? This is something I heard from someone one time. I ran across it again today in a reddit comment:

The real issue is that mean square error equates to a normal distribution over the output.

Is there any truth to this, and if so, why?

No, there isn't.  Mean square error loss is just that; a way of evaluating the errors from a model + estimation procedure in terms of a single metric. It has nothing whatsoever to do with probability distributions, except for the coincidental fact that minimizing MSE is equivalent to maximizing the likelihood if the errors actually do follow a Normal distribution.
As a working example, where both the independent variable and the errors have Exponential distributions but we are still minimizing MSE:
x <- rexp(100)
y <- x + rexp(100)
m0 <- lm(y~x)
hist(residuals(m0))
hist(fitted(m0))

gives the following two histograms:


... and you can see that neither has anything near a Normal distribution.
