Hydrogen spillover on carbon-based systems has been proposed as a viable alternative for room-temperature storage. Given the strength of the C-H bonds, however, it is unclear if spillover indeed takes place in such materials. We performed a first-principles study of H spillover on IRMOF-1. Spillover becomes thermodynamically stable only at high H coverage with a calculated Gibbs free energy of -14 kJ/mol at ambient condition. In general, however, spillover may not proceed due to high-energy states at lower H coverage. We propose that hole doping can substantially lower the energies as well as barriers to enable spillover at ambient conditions.