Exciton-exciton interactions are key to understanding nonlinear optical and transport phenomena in van der Waals heterobilayers, which emerged as versatile platforms to study correlated electronic states. We present a combined theory-experiment study of excitonic many-body effects based on first-principle band structures and Coulomb interaction matrix elements. Key to our approach is the explicit treatment of the fermionic substructure of excitons and dynamical screening effects for density-induced energy renormalization and dissipation. We demonstrate that dipolar blueshifts are almost perfectly compensated by many-body effects, mainly by screening-induced self-energy corrections. Moreover, we identify a crossover between attractive and repulsive behavior at elevated exciton densities. Theoretical findings are supported by experimental studies of spectrally narrow, mobile interlayer excitons in atomically reconstructed, h-BN-encapsulated MoSe2/WSe2 heterobilayers. Both theory and experiment show energy renormalization on a scale of a few meV even for high injection densities in the vicinity of the Mott transition. Our results revise the established picture of dipolar repulsion dominating exciton-exciton interactions in van der Waals heterostructures and open up opportunities for their external design.