A search is performed for a heavy particle decaying into different flavour dilepton pairs ($e\mu$, $e\tau$ or $\mu\tau$), using 3.2 $fb^{-1}$ of proton-proton collision data at $\sqrt{s}=13$ TeV collected in 2015 by the ATLAS detector at the Large Hadron Collider. No excess over the Standard Model prediction is observed. Limits at the 95% credibility level are set on the mass of a $Z^{\prime}$ boson with lepton-flavour-violating couplings at 3.0, 2.7 and 2.6 TeV, and on the mass of a supersymmetric $\tau$ sneutrino with $R$-parity-violating couplings at 2.3, 2.2 and 1.9 TeV, for $e\mu$, $e\tau$ and $\mu\tau$ final states, respectively. The results are also interpreted as limits on the threshold mass for quantum black hole production.