  We study $f(T)$ cosmological models inserting a non-vanishing spatial
curvature and discuss its consequences on cosmological dynamics. To figure this
out, a polynomial $f(T)$ model and a double torsion model are considered. We
first analyze those models with cosmic data, employing the recent surveys of
Union 2.1, baryonic acoustic oscillation and cosmic microwave background
measurements. We then emphasize that the two popular $f(T)$ models enable the
crossing of the phantom divide line due to dark torsion. Afterwards, we compute
numerical bounds up to 3-$\sigma$ confidence level, emphasizing the fact that
$\Omega_{k0}$ turns out to be non-compatible with zero at least at 1$\sigma$.
Moreover, we underline that, even increasing the accuracy, one cannot remove
the degeneracy between our models and the $\Lambda$CDM paradigm. So that, we
show that our treatments contain the concordance paradigm and we analyze the
equation of state behaviors at different redshift domains. We also take into
account gamma ray bursts and we describe the evolution of both the $f(T)$
models with high redshift data. We calibrate the gamma ray burst measurements
through small redshift surveys of data and we thus compare the main differences
between non-flat and flat $f(T)$ cosmology at different redshift ranges. We
finally match the corresponding outcomes with small redshift bounds provided by
cosmography. To do so, we analyze the deceleration parameters and their
variations, proportional to the jerk term. Even though the two models well fit
late-time data, we notice that the polynomial $f(T)$ approach provides an
effective de-Sitter phase, whereas the second $f(T)$ framework shows analogous
results compared with the $\Lambda$CDM predictions.
