In metal-organic-framework (MOF) perovskites, both magnetic and ferroelectric orderings can be readily realized by compounding spin and charge degrees of freedom. The hydrogen bonds that bridge the magnetic framework and organic molecules have long been thought of as a key in generating multiferroic properties. However, the underlying physical mechanisms remain unclear. Here, we combine neutron diffraction, quasielastic and inelastic neutron scattering, and THz spectroscopy techniques to thoroughly investigate the dynamical properties of the multiferroic MOF candidate [CH$_3$NH$_3$][Co(HCOO)$_3$] through its multiple phase transitions. The wide range of energy resolutions reachable by these techniques enables us to scrutinize the coupling between the molecules and the framework throughout the phase transitions and interrogate a possible magnetoelectric coupling. Our results also reveal a structural change around 220 K which may be associated with the activation of a nodding donkey mode of the methylammonium molecule due to the ordering of the CH$_3$ groups. Upon the occurrence of the modulated phase transition around 130 K, the methylammonium molecules undergo a freezing of its reorientational motions which is concomitant with a change of the lattice parameters and anomalies of collective lattice vibrations. No significant change has been however observed in the lattice dynamics around the magnetic ordering, which therefore indicates the absence of a substantial magneto-electric coupling in zero-field.
