  The Advanced LIGO and Advanced Virgo observatories recently discovered
gravitational waves from a binary neutron star inspiral. A short gamma-ray
burst (GRB) that followed the merger of this binary was also recorded by the
Fermi Gamma-ray Burst Monitor (Fermi-GBM), and the Anticoincidence Shield for
the Spectrometer for the International Gamma-Ray Astrophysics Laboratory
(INTEGRAL), indicating particle acceleration by the source. The precise
location of the event was determined by optical detections of emission
following the merger. We searched for high-energy neutrinos from the merger in
the GeV--EeV energy range using the ANTARES, IceCube, and Pierre Auger
Observatories. No neutrinos directionally coincident with the source were
detected within $\pm500$ s around the merger time. Additionally, no MeV
neutrino burst signal was detected coincident with the merger. We further
carried out an extended search in the direction of the source for high-energy
neutrinos within the 14-day period following the merger, but found no evidence
of emission. We used these results to probe dissipation mechanisms in
relativistic outflows driven by the binary neutron star merger. The
non-detection is consistent with model predictions of short GRBs observed at a
large off-axis angle.
