We consider a nanoelectromechanical weak link composed of a carbon nanotube suspended above a trench in a normal metal electrode and positioned in a gap between two superconducting leads. The nanotube is treated as a movable single-level quantum dot (QD) in which the position-dependent superconducting order parameter is induced as a result of Cooper pair tunneling. We show that in such a system, self-sustained bending vibrations can emerge if a bias voltage is applied between normal and superconducting electrodes. The occurrence of this effect crucially depends on the direction of the bias voltage and the relative position of the QD level. We also demonstrate that the nanotube vibrations strongly affect the dc current through the system, a characteristic that can be used for the direct experimental observation of the predicted phenomenon.