The current-induced vibrational dynamics is a key factor determining the mechanical stability of molecular nanojunctions. Beyond conventional Joule heating, a different mechanism leading to vibrational instability, which is induced by nonconservative current-induced forces, has been reported at low bias voltage. Given reports that this mechanism can induce extremely large vibrational excitation in models in which the unperturbed vibrational potentials are harmonic, a natural question is whether it can also be a source of current-induced bond rupture and dissociation in anharmonic systems. Consequently, in this work, we extend previous investigations of two-level, two-mode molecular nanojunctions to various anharmonic unperturbed vibrational potentials. Using a mixed quantum-classical approach based on electronic friction and Langevin dynamics, we first explore the impact of nonconservative forces on dissociation dynamics in Morse-type potentials. Furthermore, using quartic vibrational potentials, we use steady-state vibrational excitation and electric current to explore the impact of anharmonicity on this mechanism of vibrational instability, investigating whether it survives beyond the purely harmonic limit in this system