Optical resonators are a powerful platform to control the spontaneous emission dynamics of excitons in solid-state nanostructures. We study a ${\mathrm{MoSe}}_2\text{-}{\mathrm{WSe}}_2$ heterostructure that is integrated in a cryogenic open optical microcavity to gain insights into fundamental optical properties of the emergent interlayer excitons. First, we utilize a low-quality-factor planar open cavity and investigate the modification of the excitonic lifetime as on- and off-resonance conditions are met with consecutive longitudinal modes. Time-resolved photoluminescence measurements revealed a periodic tuning of the interlayer exciton lifetime by 220 ps, which allows us to extract a 0.5 ns free-space radiative lifetime and a quantum efficiency as high as $81.4\%\pm 1.4\%$. We subsequently engineer the local density of optical states by spatially confined and spectrally tunable Tamm-plasmon resonances. The dramatic redistribution of the local optical modes allows us to encounter a significant inhibition of the excitonic spontaneous emission rate by a factor of 3.2. Our open cavity is able to tune the cavity resonances accurately to the emitters to have a robust in situ control of the light-matter coupling. Such a powerful characterization approach can be universally applied to tune the exciton dynamics and measure the quantum efficiencies of more complex van der Waals heterostructures and devices.