If the polarity of the solvent increases, will the Rf value increase or decrease?
I wish we would stop teaching chromatography in terms of "polar" and "nonpolar." 

The aspirin will interact fairly strongly with the silica due to hydrogen bonding/electrostatic interactions of the carboxylic acid and the ester with the silica. If you increased the "polar" component of the mobile phase, it would travel further due to the mobile phase competing more for binding spots of the silica gel.

So increasing the polarity of the mobile phase does raise $\mathrm{R_f}$ of aspirin (or anything) on regular silica gel.