The presence of a population of point sources in a data set modifies the underlying neutrino-count statistics from the Poisson distribution. This deviation can be exactly quantified using the non-Poissonian template fitting technique, and in this work we present the first application of this approach to the IceCube high-energy neutrino data set. Using this method, we search in 7 yr of IceCube data for point-source populations correlated with the disk of the Milky Way, the Fermi bubbles, the Schlegel, Finkbeiner, and Davis dust map, or with the isotropic extragalactic sky. No evidence for such a population is found in the data using this technique, and in the absence of a signal, we establish constraints on population models with source-count distribution functions that can be described by a power law with a single break. The derived limits can be interpreted in the context of many possible source classes. In order to enhance the flexibility of the results, we publish the full posterior from our analysis, which can be used to establish limits on specific population models that would contribute to the observed IceCube neutrino flux.