A search for pair production of heavy scalar leptoquarks (LQs), each decaying into a top quark and a $\tau$ lepton, is presented. The search considers final states with an electron or a muon, one or two $\tau$ leptons that decayed to hadrons, and additional jets. The data were collected in 2016 in proton-proton collisions at $\sqrt{s}=$ 13 TeV with the CMS detector at the LHC, and correspond to an integrated luminosity of 35.9 fb$^{-1}$. No evidence for pair production of LQs is found. Assuming a branching fraction of unity for the decay LQ$ \to $t$\tau$, upper limits on the production cross section are set as a function of LQ mass, excluding masses below 900 GeV at 95% confidence level. These results provide the most stringent limits to date on the production of scalar LQs that decay to a top quark and a $\tau$ lepton.