Next-to-leading-order (NLO) quantum electrodynamics (QED) corrections to the production of muon and tau pairs in photon-photon collisions, $\gamma\gamma\to\mu^{+}\mu^{-},\tau^{+}\tau^{-}$, are calculated in the equivalent photon approximation. We mostly consider $\gamma\gamma$ processes in ultraperipheral collisions of hadrons at the LHC, but the $\gamma\gamma\to\tau^{+}\tau^{-}$ process in $\mathrm{e}^+\mathrm{e}^-$ collisions at LEP is also discussed. The NLO terms are found to modify the total cross sections by up to 5%, increasing the tails of the dilepton acoplanarity and transverse momentum distributions, and depleting by up to 15% the yields at high masses, with respect to the leading-order predictions including the very small virtuality of the colliding photons. At the LHC, the calculations obtained with the charge form factor for protons and lead ions including the NLO QED corrections improve the data--theory agreement for all measured differential distributions, and prove an indispensable ingredient for the extraction of precision quantities in photon-photon processes, such as the anomalous magnetic moment of the tau lepton.