A search for physics beyond the standard model (SM) in the final state with a hadronically decaying tau lepton and a neutrino is presented. This analysis is based on data recorded by the CMS experiment from proton-proton collisions at a center-of-mass energy of 13\TeV at the LHC, corresponding to a total integrated luminosity of 138~\fbinv. No significant deviation from the SM prediction is observed. Limits are set on the production cross section of a \Wprime boson decaying into tau lepton plus neutrino. Lower limits are set on the mass of the sequential SM-like heavy charged vector boson and the mass of a quantum black hole. Upper limits are placed on the couplings of a new boson to the SM fermions. Constraints are put on a nonuniversal gauge interaction model and an effective field theory model. For the first time, upper limits on the cross section of $t$-channel leptoquark (LQ) exchange are presented and are translated into exclusion limits on the mass and coupling of LQs in this search channel; the sensitivity of this analysis reaches into the parameter space of LQ models that attempt to explain the anomalies observed in B-meson decays. The limits presented for the various interpretations are the most stringent to date.