The first measurement of the high-mass $\tau\bar{\tau}$ production cross-section is presented, using $140~\mathrm{fb}^{-1}$ of proton-proton collision data, recorded with the ATLAS detector at $\sqrt{s} = 13$ TeV. The measurement agrees with the predictions of the Standard Model. The same process is also exploited to constrain new physics models affecting the Standard Model flavour sector. A fit to the $\tau$-lepton pair invariant mass distribution is performed as a function of $b$-jet multiplicity, to constrain the non-resonant production of leptoquark particles and heavy $Z'$ bosons that couple preferentially to third generation leptons, as well as new interactions described by an effective field theory. The constraints on leptoquark particles improve on previous results and exclude models preferred to interpret the flavour anomalies in $B$-hadron decays. The constraints on effective field theory couplings include those affecting $g-2$ of the $\tau$-lepton.