We report experimental evidence of thermal effects on the displacement of vortex walls in NiFe nanostrips. With the use of nanosecond current pulses, a unidirectional motion of the magnetic domain walls towards the hotter part of the nanostrips is observed, in addition to current-induced domain wall motion. By tuning the heat dissipation in the samples and modeling the heat diffusion, we conclude that this unidirectional motion can only be explained by the presence of a temperature profile along the nanostrip. A quantitative analysis of the experiments shows that, on top of the classical thermodynamic pressure on the domain wall, another force, probably the magnonic spin Seebeck effect, is displacing the domain walls.