Well defined magnetic excitations in holmium trifluoride are observed in inelastic neutron-scattering experiments, both in the paramagnetic phase at 1.6 K and in the ferrimagnetic phase at 90 mK. The dispersion relations at the two temperatures have been determined along the high-symmetry directions, and the field dependence of the excitations has been studied at 1.6 K. These measurements and the previous studies of the magnetic properties of HoF₃ are analysed within the mean-field/random-phase approximation (RPA). The two lowest electronic states of Ho³⁺ ions are singlets, which are well separated from the remaining levels. The dominating coupling between the ions is the classical dipole interaction, which forces the system to order at T_c=0.53 K. The two-ion coupling is below the threshold value for inducing the ordering of the electronic system; it only occurs because the magnetic susceptibility is enhanced by the hyperfine interaction between the electronic and nuclear moments on the Ho ions. The classical dipole coupling is calculated directly from first principles, whereby the response function is nearly fixed by the macroscopic properties of the system. The calculated response is found to agree accurately with the observations in the paramagnetic phase, whereas some discrepancies occur in the ordered phase. These may indicate that correlation effects beyond the RPA are important or that two-ion (magnetoelastic) quadrupole couplings are present.