We present sensitive radio continuum imaging at 1.4 and 4.9 GHz of seven high-redshift QSOs selected for having a 240 GHz continuum detection, which is thought to be thermal dust emission. We detect radio continuum emission from four of the sources: BRI 0952-0115, BR 1202-0725, LBQS 1230+1627B, and BRI 1335-0417. The radio source in BR 1202-0725 is resolved into two components, coincident with the double millimeter and CO sources. We compare the results at 1.4 and 240 GHz to empirical and semianalytic spectral models based on star-forming galaxies at low redshift. The radio-to-submillimeter spectral energy distribution (SED) for BR 1202-0725, LBQS 1230+1627B, and BRI 1335-0417 are consistent with that expected for a massive starburst galaxy, with implied massive star formation rates of order 103 Msun yr-1 (without correcting for possible amplification by gravitational lensing). The radio-to-submillimeter SED for BRI 0952-0115 suggests a low-luminosity radio jet source driven by the active galactic nucleus.