We study experimentally the dissipative dynamics of ultracold bosonic gases in a dynamic disorder potential with tunable correlation time. First, we measure the heating rate of thermal clouds exposed to the dynamic potential and present a model of the heating process, revealing the microscopic origin of dissipation from a thermal, trapped cloud of bosons. Second, for Bose-Einstein condensates, we measure the particle loss rate induced by the dynamic environment. Depending on the correlation time, the losses are either dominated by heating of residual thermal particles or the creation of excitations in the superfluid, a notion we substantiate with a rate model. Our results illuminate the interplay between superfluidity and time-dependent disorder and on more general grounds establish ultracold atoms as a platform for studying spatiotemporal noise and time-dependent disorder.