We have studied the temperature dependence and anharmonic coupling of the local vibrational modes (LVMs) associated with Mg-H complexes in heavily doped InN:Mg. Two main LVM peaks are observed which are probably related to two different H-impurity bond lengths. The temperature dependence of the higher-frequency mode, which exhibits a monotonic frequency downshift and broadening with increasing temperature, can be explained by LVM dephasing due to acoustic phonon scattering. The lower-frequency mode displays an anomalous behavior as its frequency decreases initially and then starts to increase linearly above room temperature. The anharmonic coupling of the lower-frequency mode to a molecular mode of the impurity complex is suggested as a possible cause for this behavior.