Nanocrystalline InP quantum dots (QDs) hold promise for heavy-metal-free optoelectronic applications due to their bright and size-tunable emission in the visible range. Photochemical stability and high photoluminescence (PL) quantum yield are obtained by a diversity of epitaxial shells around the InP core. To understand and optimize the emission line shapes, the exciton fine structure of InP core/shell QD systems needs be investigated. Here, we study the exciton fine structure of InP/ZnSe core/shell QDs with core diameters ranging from 2.9 to 3.6 nm (PL peak from 2.3 to 1.95 eV at 4 K). PL decay measurements as a function of temperature in the 10 mK to 300 K range show that the lowest exciton fine structure state is a dark state, from which radiative recombination is assisted by coupling to confined acoustic phonons with energies ranging from 4 to 7 meV, depending on the core diameter. Circularly polarized fluorescence line-narrowing (FLN) spectroscopy at 4 K under high magnetic fields (up to 30 T) demonstrates that radiative recombination from the dark F = ±2 state involves acoustic and optical phonons, from both the InP core and the ZnSe shell. Our data indicate that the highest intensity FLN peak is an acoustic phonon replica rather than a zero-phonon line, implying that the energy separation observed between the F = ±1 state and the highest intensity peak in the FLN spectra (6 to 16 meV, depending on the InP core size) is larger than the splitting between the dark and bright fine structure exciton states.