We present a search for coincidence between IceCube TeV neutrinos and fast radio bursts (FRBs). During the search period from 2010 May 31 to 2016 May 12, a total of 29 FRBs with 13 unique locations have been detected in the whole sky. An unbinned maximum likelihood method was used to search for spatial and temporal coincidence between neutrinos and FRBs in expanding time windows, in both the northern and southern hemispheres. No significant correlation was found in six years of IceCube data. Therefore, we set upper limits on neutrino fluence emitted by FRBs as a function of time window duration. We set the most stringent limit obtained to date on neutrino fluence from FRBs with an E⁻² energy spectrum assumed, which is 0.0021 GeV cm⁻² per burst for emission timescales up to ~10² s from the northern hemisphere stacking search.