Decades of searches for Population III stars in individual galaxies have yielded a few potential candidates, but a statistically robust characterization of the demographics of the first stars in the lowest-mass systems remains elusive. Line intensity mapping (LIM), an observational technique that measures fluctuations in the aggregate emission from the entire galaxy population — including the faintest sources — offers an alternative strategy that is especially well-suited for the Pop III era. With the recent launch of SPHEREx and the rapid development of a number of complementary LIM studies, we are poised to place some of the first LIM constraints on sites of star formation at high-redshift. In this work, we expand an analytical model for LIM power spectra, Zeus21/oLIMpus, to include Pop III stars and the emission lines identified as diagnostic signatures of star formation with a low-metallicity, top-heavy IMF, such as Hα and HeII. We introduce a flexible framework to estimate the measurement uncertainties associated with instrument and survey configurations, and apply these to study LIM signatures of the first stars in mock surveys carried out with SPHEREx and potential next-generation instruments. We quantify the sensitivity of the LIM signal to variations in Pop II and III parameters and forecast joint limits that can be placed on the Pop III star formation efficiency (SFE) and IMF shape with SPHEREx. We find that next-generation instruments will be necessary to comprehensively survey the Pop III theoretical landscape — both with respect to classical and exotic Pop III models — and identify design improvements that will enable such studies. Finally, we carry out a Fisher analysis to characterize synergies between SPHEREx and JWST, suggesting that joint constraints on the Pop III SFE and extensions to conventional Pop III models may be within reach.