Van der Waals heterostructures based on transition metal dichalcogenides (TMDs) have emerged as excellent candidates for next-generation optoelectronics and valleytronics, due to their fascinating physical properties. The understanding and active control of the relaxation dynamics of heterostructures play a crucial role in device design and optimization. Here, we investigate the back-gate modulation of exciton dynamics in a WS2/WSe2 heterostructure by combining time-resolved photoluminescence (TRPL) and transient absorption spectroscopy (TAS) at cryogenic temperatures. We find that the non-radiative relaxation lifetimes of photocarriers in heterostructures can be electrically controlled for samples with different twist-angles, whereas such lifetime tuning is not present in standalone monolayers. We attribute such an observation to doping-controlled competition between interlayer and intralayer recombination pathways in high-quality WS2/WSe2 samples. The simultaneous measurement of TRPL and TAS lifetimes within the same sample provides additional insight into the influence of coexisting excitons and background carriers on the photo-response, and points to the potential of tailoring light-matter interactions in TMD heterostructures.