This article presents searches for the $Z\gamma$ decay of the Higgs boson and for narrow high-mass resonances decaying to $Z\gamma$, exploiting $Z$ boson decays to pairs of electrons or muons. The data analysis uses 36.1 fb$^{-1}$ of $pp$ collisions at $\sqrt{s} = 13$ TeV recorded by the ATLAS detector at the CERN Large Hadron Collider. The data are found to be consistent with the expected Standard Model background. The observed (expected - assuming Standard Model $pp\to H\to Z\gamma$ production and decay) upper limit on the production cross section times the branching ratio for $pp\to H\to Z\gamma$ is 6.6 (5.2) times the Standard Model prediction at the 95% confidence level for a Higgs boson mass of 125.09 GeV. In addition, upper limits are set on the production cross section times the branching ratio as a function of the mass of a narrow resonance between 250 GeV and 2.4 TeV, assuming spin-0 resonances produced via gluon-gluon fusion, and spin-2 resonances produced via gluon-gluon or quark-antiquark initial states. For high-mass spin-0 resonances, the observed (expected) limits vary between 88 fb (61 fb) and 2.8 fb (2.7 fb) for the mass range from 250 GeV to 2.4 TeV at the 95% confidence level.