In pursuit of ambitious decarbonization goals, the global energy market is rapidly transitioning from conventional energy sources to sustainable alternatives that mitigate carbon emissions. Green hydrogen, derived from renewable energy sources, emerges as a promising candidate for production, transportation to high-demand regions, and subsequent storage to meet energy demand fluctuations. However, conventional tank storage for hydrogen presents operational challenges, requiring complex operational conditions to cope with low temperatures and high pressures. Underground hydrogen storage within depleted reservoirs has economic advantages, as it capitalizes on the subsurface environment and on existing pipeline infrastructure. The cycles of injection and extraction of hydrogen into/from the subsurface induce a pressure build-up within the porous storage formation, consequently reducing effective stress and potentially influencing fault stability. In this study, the geomechanical implications of pore pressure accumulation resulting from seasonal hydrogen storage are thoroughly investigated. Notably, hydrogen injection near a low-permeability fault induces pressurization within the storage formation between the injection well and the fault. The restricted fluid flow across the fault leads to diminished overpressure on the opposite side, creating a spatial variability in fluid pressure distribution that contributes to total stress changes around the fault, ultimately diminishing its stability. Engineering appropriate operating pressures, injection/recovery rates, and total injected volumes emerges as a critical factor in ensuring the safety of the storage operation.