We present two searches for IceCube neutrino events coincident with 28 fast radio bursts (FRBs) and 1 repeating FRB. The first improves on a previous IceCube analysis-searching for spatial and temporal correlation of events with FRBs at energies greater than roughly 50 GeV-by increasing the effective area by an order of magnitude. The second is a search for temporal correlation of MeV neutrino events with FRBs. No significant correlation is found in either search; therefore, we set upper limits on the time-integrated neutrino flux emitted by FRBs for a range of emission timescales less than one day. These are the first limits on FRB neutrino emission at the MeV scale, and the limits set at higher energies are an order-of-magnitude improvement over those set by any neutrino telescope.