Hyperferroelectrics are observing a growing interest thanks to their unique property to retain a spontaneous polarization even in presence of a depolarizing field, corresponding to zero macroscopic displacement field (D = 0) conditions. Hyperferroelectricity is ascribed to the softening of a polar LO mode, but the microscopic mechanisms behind this softening are not totally resolved. Here, by means of phonon calculations and force constants analysis, performed in two class of hyperferroelectrics, the ABO3-LiNbO3-type systems and the hexagonal-ABC systems, we unveiled the common features in the dynamical properties of a hyperferroelectric that are leading the LO instability: negative or vanishing on-site force constant associated to the cation driving the LO polar mode and a destabilizing cation-anion interactions; both induced by short-range forces. We also predicted a possible enhancement of the hyperferroelectric properties under increasing external positive pressures: the pressure strengthens the destabilizing short-range interactions, inducing a stronger LO mode instability and the increase of the longitudinal mode effective charges associated to the unstable LO mode, suggesting an eventual enhancement of the D = 0 polarization, under compressive strain.