Disclination loops threaded on a polymeric fiber with planar axial anchoring are shown to be sensitive to magnetic fields. In particular, a homogeneous magnetic field oblique to the fiber drives translation of disclination loops along it. The velocity of translation depends on the intensity of the field and on the angle between the field and the fiber. It is shown experimentally that an inhomogeneous magnetic field of an appropriate geometry can act as a trap for disclination loops. When such a magnetic trap is translated along the fiber, it drags the loop with it and therefore acts as magnetic tweezers. Theoretical model of the translational stability of disclination loops in magnetic fields is proposed.