Many-body effects in ice are investigated through a systematic analysis of the lattice energies of several proton ordered and disordered phases, which are calculated with different flexible water models, ranging from pairwise additive (q-TIP4P/F) to polarizable (TTM3-F and AMOEBA) and explicit many-body (MB-pol) potential energy functions. Comparisons with available experimental and diffusion Monte Carlo data emphasize the importance of an accurate description of the individual terms of the many-body expansion of the interaction energy between water molecules for the correct prediction of the energy ordering of the ice phases. Further analysis of the MB-pol results, in terms of fundamental energy contributions, demonstrates that the differences in lattice energies between different ice phases are sensitively dependent on the subtle balance between short-range two-body and three-body interactions, many-body induction, and dispersion energy. By correctly reproducing many-body effects at both short range and long range, it is found that MB-pol accurately predicts the energetics of different ice phases, which provides further support for the accuracy of MB-pol in representing the properties of water from the gas to the condensed phase.