  We use observations related to the variation of fundamental constants, in
order to impose constraints on the viable and most used $f(T)$ gravity models.
In particular, for the fine-structure constant we use direct measurements
obtained by different spectrographic methods, while for the effective Newton's
constant we use a model-dependent reconstruction, using direct observational
Hubble parameter data, in order to investigate its temporal evolution. We
consider two $f(T)$ models and we quantify their deviation from $\Lambda$CDM
cosmology through a sole parameter. Our analysis reveals that this parameter
can be slightly different from its $\Lambda$CDM value, however the best-fit
value is very close to the $\Lambda$CDM one. Hence, $f(T)$ gravity is
consistent with observations, nevertheless, as every modified gravity, it may
exhibit only small deviations from $\Lambda$CDM cosmology, a feature that must
be taken into account in any $f(T)$ model-building.
