We present the Fermi Large Area Telescope (LAT) observations of the binary neutron star merger event GW170817 and the associated short gamma-ray burst (SGRB) GRB 170817A detected by the Fermi Gamma-ray Burst Monitor. The LAT was entering the South Atlantic Anomaly at the time of the LIGO/Virgo trigger (t GW) and therefore cannot place constraints on the existence of high-energy (E > 100 MeV) emission associated with the moment of binary coalescence. We focus instead on constraining high-energy emission on longer timescales. No candidate electromagnetic counterpart was detected by the LAT on timescales of minutes, hours, or days after the LIGO/Virgo detection. The resulting flux upper bound (at 95% C.L.) from the LAT is 4.5 × 10−10 erg cm−2 s−1 in the 0.1–1 GeV range covering a period from t GW + 1153 s to t GW + 2027 s. At the distance of GRB 170817A, this flux upper bound corresponds to a luminosity upper bound of 9.7 × 1043 erg s−1, which is five orders of magnitude less luminous than the only other LAT SGRB with known redshift, GRB 090510. We also discuss the prospects for LAT detection of electromagnetic counterparts to future gravitational-wave events from Advanced LIGO/Virgo in the context of GW170817/GRB 170817A.