We have made the first investigation of the external magnetic field effect on director dynamics and flow instability in liquid-crystalline polymers subjected to shear flow by Brownian dynamics simulation and Lebwohl-Lasher nematogen model. According to our simulation, the static magnetic field can, depending on the field strength and orientation, enhance or inhibit the flow instability. The corresponding state diagram for director tumbling and shear aligning is presented. The tumbling period in a magnetic field does not obey the conventional scaling relation. In addition, a novel phenomenon about the first normal stress difference vs shear rate has been found in a magnetic held.