The ability to support frictionless motion is one of the manifestations of superfluidity. An impurity immersed in a superfluid can move without dissipation below the critical velocity, which, according to the Landau criterion, is determined by the elementary excitation spectrum of the system. In a quantum gas of the ultracold atoms, the critical velocity can be measured by stirring a laser beam through the atomic cloud, and the emergence of dissipation can be observed via the heating effect above the threshold stirring speed. Recently, such a technique is exploited to study the superfluidity of the Bose-Einstein condensate (BEC) of ¹⁶²Dy atoms with dipole-dipole interactions. It is shown that both the critical velocity and the heating rate reflect the anisotropy of the underlying dipolar excitation spectrum. In this work, we theoretically investigate the anisotropic dissipation of a point-like impurity moving through a dipolar BEC. For the motion along the principal axis, the dissipation rate above the critical velocity is analytically derived according to the linear response theory. At a given reduced velocity, we find the dissipation rate being of a higher value in the direction parallel to the dipole moment, which qualitatively explains the recent experimental observation in dysprosium atoms. Moreover, in the moving direction away from the principal axis, the asymptotic expressions for the dissipation rate are obtained in the high-speed limit, as well as in the regime close to the dissipation threshold. By combining these analytical results with the numerical calculations, we conclude that, in a dipolar BEC, the angular dependence of the dissipation rate always shows the same anisotropy as the critical velocity. Our predictions can be examined in the current experiments with cold atoms, and the results presented here may be also helpful in understanding the anisotropic superfluidity in other systems.