Heterostructures made from atomically thin semiconductors (here MoSe₂) and graphene are uniquely poised to investigate photoinduced charge and energy transfer in the 2D limit. Here, using picosecond time-resolved photoluminescence spectroscopy at cryogenic temperatures on two types of MoSe₂/graphene heterostrutures, we unveil key features of the underlying mechanisms. First, the shortening of the MoSe₂ bright exciton lifetime is marginally affected by the number of graphene layers to which MoSe₂ is coupled. Second, exciton transfer vanishes when a sub-nm thick spacer of hexagonal boron nitride decouples MoSe₂ from graphene. These results indicate that charge tunneling govern bright exciton relaxation in MoSe₂/graphene and that longer-range, Förster-type energy transfer (FRET) does not affect bright excitons. However, sub-ps FRET to graphene accelerates the relaxation of ``hot'' excitons formed upon optical excitation, leading to photoluminescence quenching factors that exceed expectations based on the shortening of the bright exciton lifetime. Our work has direct implications for energy harvesting and funneling using van der Waals heterostructures.