The majority of the ultimate main-sequence mass of a star is assembled in the protostellar phase, where a forming star is embedded in an infalling envelope and encircled by a protoplanetary disk. Studying mass accretion in protostars is thus a key to understanding how stars gain their mass and ultimately how their disks and planets form and evolve. At this early stage, the dense envelope reprocesses most of the luminosity generated by accretion to far-infrared and submillimeter wavelengths. Time-domain photometry at these wavelengths is needed to probe the physics of accretion onto protostars, but variability studies have so far been limited, in large part because of the difficulty in accessing these wavelengths from the ground. We discuss the scientific progress that would be enabled with far-infrared and submillimeter programs to probe protostellar variability in the nearest kiloparsec.