The first observation of a binary neutron star (NS) coalescence by the Advanced LIGO and Advanced Virgo gravitational-wave (GW) detectors offers an unprecedented opportunity to study matter under the most extreme conditions. After such a merger, a compact remnant is left over whose nature depends primarily on the masses of the inspiraling objects and on the equation of state of nuclear matter. This could be either a black hole (BH) or an NS, with the latter being either long-lived or too massive for stability implying delayed collapse to a BH. Here, we present a search for GWs from the remnant of the binary NS merger GW170817 using data from Advanced LIGO and Advanced Virgo. We search for short- (≲1 s) and intermediate-duration (≲500 s) signals, which include GW emission from a hypermassive NS or supramassive NS, respectively. We find no signal from the post-merger remnant. Our derived strain upper limits are more than an order of magnitude larger than those predicted by most models. For short signals, our best upper limit on the root sum square of the GW strain emitted from 1-4 kHz is at 50% detection efficiency. For intermediate-duration signals, our best upper limit at 50% detection efficiency is for a millisecond magnetar model, and for a bar-mode model. These results indicate that post-merger emission from a similar event may be detectable when advanced detectors reach design sensitivity or with next-generation detectors.