GaSb-rich superlattice (SL) p-i-n photodiodes grown by molecular beam epitaxy were studied theoretically and experimentally in order to understand the poor dark current characteristics typically obtained. This behavior, independent of the SL-grown material quality, is usually attributed to the presence of defects due to Ga-related bonds, limiting the SL carrier lifetime. By analyzing the photoresponse spectra of reverse-biased photodiodes at 80 K, we have highlighted the presence of an electric field, breaking the minibands into localized Wannier-Stark states. Besides the influence of defects in such GaSb-rich SL structures, this electric field induces a strong tunneling current at low bias which can be the main limiting mechanism explaining the high dark current density of the GaSb-rich SL diode.