Radio emission from radio-quiet quasars may be due to star formation in the quasar host galaxy, to a jet launched by the supermassive black hole, or to relativistic particles accelerated in a wide-angle radiatively driven outflow. In this paper, we examine whether radio emission from radio-quiet quasars is a byproduct of star formation in their hosts. To this end, we use infrared spectroscopy and photometry from Spitzer and Herschel to estimate or place upper limits on star formation rates in hosts of ~300 obscured and unobscured quasars at z<1. We find that low-ionization forbidden emission lines such as [NeII] and [NeIII] are likely dominated by quasar ionization and do not provide reliable star formation diagnostics in quasar hosts, while polycyclic aromatic hydrocarbon (PAH) emission features may be suppressed due to the destruction of PAH molecules by the quasar radiation field. While the bolometric luminosities of our sources are dominated by the quasars, the 160{mu}m fluxes are likely dominated by star formation, but they too should be used with caution. We estimate median star formation rates to be 6-29M_{sun}_/yr, with obscured quasars at the high end of this range. This star formation rate is insufficient to explain the observed radio emission from quasars by an order of magnitude, with log(L_radio,obs_/L_radio,SF_)=0.6-1.3 depending on quasar type and star formation estimator. Although radio-quiet quasars in our sample lie close to the 8-1000{mu}m infrared/radio correlation characteristic of the star-forming galaxies, both their infrared emission and their radio emission are dominated by the quasar activity, not by the host galaxy.