A planar metal-dielectric multilayer structure containing a waveguide layer doped with azo-dye molecules exhibits a sharp Fano resonance that can be shifted by pump light irradiation owing to a photoinduced change of the refractive index of the waveguide layer by photoisomerization of the azo-dye molecules. Azo-dye-containing materials are polarization sensitive and the photoinduced shift of the Fano resonance is expected to depend on the pump polarization. Previous pump-probe attenuated-total-reflection measurements performed with a normal incidence of pump light exhibited intriguingly no dependence on the pump polarization. In this work, we extend our measurements to an oblique incidence, to optimize the overlap between the pump and probe polarizations, and we demonstrate a strong dependence on the pump polarization. To analyze the experimental results for both the normal and oblique pump conditions, we introduce phenomenological equations, based on the angular hole burning model of the photoisomerization of the azo-dye molecules. The equations allow us to describe the dependence of the refractive index changes in the waveguide layer on the pump angle and polarization. Using the equations, we could obtain good theoretical fits to the experimental data. The results of fitting suggest that the overlap between the pump and probe polarizations plays an important role in determining the amount of the photoinduced shift of the Fano resonance.