  In analogy to $f(R)$ theory, recently a new modified gravity theory, namely
the so-called $f(T)$ theory, has been proposed to drive the current accelerated
expansion without invoking dark energy. In the present work, by extending
Bisabr's idea, we try to constrain $f(T)$ theories with the varying fine
structure "constant", $\alpha\equiv e^2/\hbar c$. We find that the constraints
on $f(T)$ theories from the observational $\Delta\alpha/\alpha$ data are very
severe. In fact, they make $f(T)$ theories almost indistinguishable from
$\Lambda$CDM model.
