Normality of errors is needed specifically for the $F$-test's exact null distribution

Consider the linear model $\underline y_{n-1}=X\underline\theta_{p-1} +\underline\xi_{n-1}$ where $E(\xi)=\underline0,\ \cov(\xi)=\sigma^2I_n$. The assumption that $\xi$ is multivariate Normal is needed for: