Red quasars are a population, characterized by significant extinction in UV, which could be explained by absorption of dusty gas on a scale of a few kpc. We show that the enhanced radiation-pressure drives the dusty gas to supersonically expand and produces shocks. The shocks energize electrons to be relativistic via the first Fermi acceleration. As a balance result of shock acceleration and synchrotron emission and inverse Compton scattering of the electrons, the maximum of the Lorentz factor of the electrons reaches as $\sim 10^6$. Synchrotron emission from the electrons peaks at near infrared band and inverse Compton scattering around 1.0GeV$-$0.1TeV. Future images of the extended multiwavelength fuzz in radio, infrared and $\gamma-$rays would provide new clues to study the details of radiative feedback if red quasars could be a certain phase in evolutionary chains of galaxies.