  We obtain $\Omega$-results for linear exponential sums with rational additive
twists of small prime denominators weighted by Hecke eigenvalues of Maass cusp
forms for the group $\mathrm{SL}_3(\mathbb Z)$. In particular, our
$\Omega$-results match the expected conjectural upper bounds when the
denominator of the twist is sufficiently small compared to the length of the
sum. Non-trivial $\Omega$-results for sums over short segments are also
obtained. Along the way we produce lower bounds for mean squares of the
exponential sums in question and also improve the best known upper bound for
these sums in some ranges of parameters.
