We analyze the deviations from local thermodynamic equilibrium (LTE) of the ion velocity distribution function (iVDF) in collisionless plasma turbulence. Using data from the magnetospheric multiscale (MMS) mission, we examine the non-Maxwellianity of 439 685 iVDFs in the Earth's magnetosheath. We find that the iVDFs' anisotropies and the high-order non-bi-Maxwellian features are widespread and can be significant. Our results show that the complexity of the iVDFs is strongly influenced by the ion plasma beta and turbulence intensity, with high-order non-LTE features emerging in the presence of large-amplitude magnetic field fluctuations. Furthermore, our analysis indicates that turbulence-driven magnetic curvature contributes to the isotropization of the iVDFs by scattering the ions, emphasizing the complex interaction between turbulence and the velocity distribution of charged particles in collisionless plasmas.