We consider a class of gauge glass models with Gaussian disorder on the Nishimori line, including the Ising spin glass, the $XY$ gauge glass, the $Z_q$ gauge glass, and the gauge-invariant Potts model. We prove that the first and second Griffiths inequalities hold for these models on arbitrary lattice structures. As a consequence, both the pressure and the correlation functions are monotonically increasing with respect to the inverse temperature along the Nishimori line. Furthermore, we establish an analogue of the Gibbs--Bogoliubov inequality for this class of models. This result implies that, on the Nishimori line, the approximate quenched free energy obtained via the replica method with a replica-symmetric mean-field approximation is always greater than the true quenched free energy. Our results provide a broad generalization of previous results established for the Ising spin glass with Gaussian disorder on the Nishimori line.