A search for neutrino and antineutrino events correlated with 2350 gamma-ray bursts (GRBs) is performed with Borexino data collected between December 2007 and November 2015. No statistically significant excess over background is observed. We look for electron antineutrinos (View the MathML source) that inverse beta decay on protons with energies from 1.8 MeV to 15 MeV and set the best limit on the neutrino fluence from GRBs below 8 MeV. The signals from neutrinos and antineutrinos from GRBs that scatter on electrons are also searched for, a detection channel made possible by the particularly radio-pure scintillator of Borexino. We obtain currently the best limits on the neutrino fluence of all flavors and species below 7 MeV. Finally, time correlations between GRBs and bursts of events are investigated. Our analysis combines two semi-independent data acquisition systems for the first time: the primary Borexino readout optimized for solar neutrino physics up to a few MeV, and a fast waveform digitizer system tuned for events above 1 MeV.