In this paper, we report the first experimental results on capillary shear flows of a nematic liquid crystal 5CB (4-cyano-4′-pentylbiphenyl), arising due to interaction of the anisotropic liquid, correspondent to the continuous rotational symmetry, with photo-profiled polymer surfaces. The regular surface relief was obtained due to opto-mechanical deformation of azobenzene containing potoresponsive polymer film (PAZO) during irradiation with two-beam interference. Such surface treatment makes it possible to obtain a regular submicron profile with well-defined characteristics (direction, period, and height). The polarizing microscopy (PM) and dynamic light scattering (DLS) techniques were used to determine the direction of the surface orientation of LC and anchoring strength, which characterize the interaction of LC with the photo-profiled polymer surface. Two types of shear flows—spreading of LC droplets and capillary flow in a plane capillary, induced by the interaction of LC with one or two photo-profiled surfaces—were investigated for different directions of the flow relative to the direction of the relief. Strong anisotropy in the dynamics of the precursor film and contact line motion, as well as in the dynamical contact angle, was established. The experimental results were analyzed and compared with those previously obtained at the investigation of the spreading of LC droplets over a mechanically stamped submicron profile and capillary flows in plane capillaries with photo-aligned surfaces.