Hi Tim, good question! So, it turns out the errors need not be normally distributed for the OLS estimator to be unbiased. In the case where the errors are normally distributed and i.i.d., this is the special case where MLE reduces to the OLS.
So, the OLS will return unbiased estimates of location parameters, and as long as the errors are uncorrelated and homoscedastic the OLS is also the BLUE estimator. However without the normality and i.i.d. conditions as well, the OLS will only be the BLUE but not the MLE. Meaning that in the universe of all unbiased estimators (not just additive linear estimators), the OLS will not be the unbiased estimator with minimum variance. However, it is will still unbiased.
Does that answer your question?