  Hybrid van der Waals heterostructures of organic semiconductors and
transition metal dichalcogenides (TMDs) are promising candidates for various
optoelectronic devices, such as solar cells and biosensors. Energy-transfer
processes in these materials are crucial for the efficiency of such devices,
yet they are poorly understood. In this work, we develop a fully microscopic
theory describing the effect of the F\"{o}rster interaction on exciton dynamics
and optics in a WSe$_2$/tetracene heterostack. We demonstrate that the
differential absorption and time-resolved photoluminescence can be used to
track the real-time evolution of excitons. We predict a strongly unidirectional
energy transfer from the organic to the TMD layer. Furthermore, we explore the
role temperature has in activating the F\"{o}rster transfer and find a good
agreement to previous experiments. Our results provide a blueprint to tune the
light-harvesting efficiency through temperature, molecular orientation and
interlayer separation in TMD/organic heterostructures.
