  We explore the impact of the Sandage-Loeb (SL) test on the precision of
cosmological constraints for $f(T)$ gravity theories. The SL test is an
important supplement to current cosmological observations because it measures
the redshift drift in the Lyman-$\alpha$ forest in the spectra of distant
quasars, covering the "redshift desert" of $2 \lesssim z \lesssim5$. To avoid
data inconsistency, we use the best-fit models based on current combined
observational data as fiducial models to simulate 30 mock SL test data. We
quantify the impact of these SL test data on parameter estimation for $f(T)$
gravity theories. Two typical $f(T)$ models are considered, the power-law model
$f(T)_{PL}$ and the exponential-form model $f(T)_{EXP}$. The results show that
the SL test can effectively break the existing strong degeneracy between the
present-day matter density $\Omega_m$ and the Hubble constant $H_0$ in other
cosmological observations. For the considered $f(T)$ models, a 30-year
observation of the SL test can improve the constraint precision of $\Omega_m$
and $H_0$ enormously but cannot effectively improve the constraint precision of
the model parameters.
