Vibrational properties and the temperature-induced phase transition mechanism have been studied in [NH4][Zn(HCOO)3] and [ND4][Zn(DCOO)3] metal organic frameworks by variable-temperature dielectric, IR, and Raman measurements. DFT calculations allowed proposing the detailed assignment of vibrational modes to respective motions of atoms in the unit cell. Temperature-dependent studies reveal a very weak isotopic effect on the phase transition temperature and confirm that ordering of ammonium cations plays a major role in the mechanism of the phase transition. We also present high-pressure Raman scattering studies on [ND4][Zn(DCOO)3]. The results indicate the rigidity of the formate ions and strong compressibility of the ZnO6 octahedra. They also reveal the onset of a pressure-induced phase transition at about 1.1 GPa. This transition has strong first-order character, and it is associated with a large distortion of the metal formate framework. Our data indicate the presence of at least two nonequivalent formate ions in the high-pressure structure with very different C-D bonds. The decompression experiment shows that the transition is reversible.