MeV neutrinos are produced in many astrophysical transients, such as stellar collapses and high-energy jets, where they play a role in sustaining and cooling energetic explosions. Detecting these neutrinos from sources outside the Milky Way is very difficult due to the small neutrino-nucleon cross section at MeV. Nevertheless, the non-observation of MeV neutrinos from high-energy transients may provide useful constraints on related neutrino production mechanisms where significant MeV production is expected. The IceCube Neutrino Observatory, a cubic kilometer neutrino detector operating with nearly 100\% uptime at the South Pole, is sensitive to bursts of MeV neutrinos from astrophysical sources in and beyond the Milky Way. In this work, we describe the MeV neutrino detection system of IceCube and show results from several categories of astrophysical transients.