Natural bornite (NBo), a sulfide mineral of copper and iron, is one of the main mineral raw materials for copper extraction. In this study, NBo-activated hydrogen peroxide (H2O2) and persulfate processes (PS) for the degradation of minocycline (MNC) in aqueous solution were systemically investigated and compared. The MNC removal rates with the NBo/PS and NBo/H2O2 processes were 86.40% and 87.50%, respectively. The mineralization rate of NBo/PS (33.96%) was higher than that of NBo/H2O2 (29.94%) after reaction for 180 min. The effects of oxidant and activator dosage, pH, common inorganic anions (i.e., Cl−, NO3−, and HCO3−), and water composition on MNC degradation were systematically evaluated. In addition, the degradation of MNC in natural water matrix and toxicity evaluation were also studied to better evaluate the feasibility of practical application of those two processes. The results of free radical quenching experiments and electron paramagnetic resonance spectroscopy (EPR) showed that HO· was the main activated species in the NBo/H2O2 system, while SO4·− and HO· were the main activated species in the NBo/PS system. The degradation of MNC in the NBo/PS system was achieved through electron transfer, while the degradation of MNC in the NBo/H2O2 system was mainly achieved through free radical addition. The degradation pathway mainly included deamidation reactions, C-C bond breakage and hydroxylation. Reusability of NBo showed that NBo was considerably stable in activating PS and H2O2 for degradation of MNC, which was cost-effective activator. This work provides a new perspective on the degradation mechanism of pollutants by Fe-Cu bimetallic sulfide activation of PS and H2O2.