  Optical resonators are a powerful platform to control the spontaneous
emission dynamics of excitons in solid-state nanostructures. Here, we study a
MoSe$_2$-WSe$_2$ van-der-Waals heterostructure that is integrated in a widely
tunable open optical microcavity to gain insights into fundamental optical
properties of the emergent interlayer charge-transfer excitons. First, we
utilize an ultra-low quality factor open planar vertical cavity and investigate
the modification of the excitonic lifetime as on- and off-resonant conditions
are met with consecutive longitudinal modes. Time-resolved photoluminescence
measurements reveal that the interlayer exciton lifetime can thus be
periodically tuned with an amplitude of 110 ps. The resulting oscillations of
the interlayer exciton lifetime allows us to extract a 0.5 ns free-space
radiative lifetime and a quantum efficiency as high as 81 \%. We subsequently
engineer the local density of optical states by introducing a spatially
confined and fully spectrally tunable Tamm-plasmon resonance. The dramatic
redistribution of the local optical modes in this setting allows us to
encounter a profound inhibition of spontaneous emission of the interlayer
excitons by a factor of 3.2. We expect that specifically engineering the
inhibition of radiation from moir\'e excitons is a powerful tool to steer their
thermalization, and eventually their condensation into coherent condensate
phases.
