As an azo dye, OG has toxic and harmful effects on ecosystems. Therefore, there is an urgent need to develop a green, environmentally friendly, and efficient catalyst to activate peroxymonosulfate (PMS) for the degradation of OG. In this study, the catalysts MIL-101(Fe) and NH2-MIL-101(Fe) were prepared using a solvothermal method to carry out degradation experiments. They were characterized by means of XRD, SEM, XPS, and FT-IR, and the results showed that the catalysts were successfully prepared. Then, a catalyst/PMS system was constructed, and the effects of different reaction systems, initial pH, temperature, catalyst dosing, PMS concentration, and the anion effect on the degradation of OG were investigated. Under specific conditions (100 mL OG solution with a concentration of 50 mg/L, pH = 7.3, temperature = 25 °C, 1 mL PMS solution with a concentration of 100 mmol/L, and a catalyst dosage of 0.02 g), the degradation of OG with MIL-101(Fe) was only 36.6% within 60 min; as a comparison, NH2-MIL-101(Fe) could reach up to 97.9%, with a reaction constant k value of 0.07245 min−1. The NH2-MIL-101 (Fe)/PMS reaction system was able to achieve efficient degradation of OG at different pH values (pH = 3~9). The degradation mechanism was analyzed using free-radical quenching tests. The free-radical quenching tests showed that SO4•−, •OH, and 1O2 were the main active species during the degradation of OG.