We measured the temperature-dependent infrared reflectivity spectra of MnF2 between 4 and 600 K. We show that the phonon spectrum undergoes a clear renormalization at TN. The ab initio calculation we performed on this compound accurately predicts the magnitude and the direction of the changes in the phonon parameters across the antiferromagnetic transition, showing that they are mainly induced by the magnetic order. In this material, we found that the dielectric constant is mostly from phonon origin. The large change in the lattice parameters with temperature seen by x-ray diffraction as well as the A2u phonon softening below TN indicate that magnetic order induced distortions in MnF2 are compatible with the ferroelectric instabilities observed in TiO2, FeF2 and other rutile-type fluorides. This study also shows the anomalous temperature evolution of the lower energy Eu mode in the paramagnetic phase, which can be compared to that of the B1g phonon seen by Raman spectroscopy in many isostructural materials. This was interpreted as being a precursor of a phase transition from rutile to CaCl2 structure which was observed under pressure in ZnF2.