We report an experimental study on the dynamics of a thin film of polymer solution coating a vertical fiber. The liquid film has first a constant thickness and then undergoes the Rayleigh-Plateau instability which leads to the formation of sequences of drops, separated by a thin film, moving down at a constant velocity. Different polymer solutions are used, i.e. xanthan solutions and polyacrylamide (PAAm) solutions. These solutions both exhibit shear-rate dependence of the viscosity, but for PAAm solutions, there are strong normal stresses in addition of the shear-thinning effect. We characterize experimentally and separately the effects of these two non-Newtonian properties on the flow on the fiber. Thus, in the flat film observed before the emergence of the drops, only shear-thinning effect plays a role and tends to thin the film compared to the Newtonian case. The effect of the non-Newtonian rheology on the Rayleigh-Plateau instability is then investigated through the measurements of the growth rate and the wavelength of the instability. Results are in good agreement with linear stability analysis for a shear-thinning fluid. The effect of normal stress can be taken into account by considering an effective surface tension which tends to decrease the growth rate of the instability. Finally, the dependence of the morphology of the drops with the normal stress is investigated and a simplified model including the normal stress within the lubrication approximation provides good quantitative results on the shape of the drops.