Relativistic heavy-ion beams at the LHC are accompanied by a large flux of nearly-real photons, leading to a variety of photon-induced processes. This note presents a series of measurements of dilepton production from photon-fusion processes performed by the ATLAS Collaboration. Recent measurements of exclusive dielectron production in ultra-peripheral collisions are presented. These processes provide strong constraints on the nuclear-photon flux and its dependence on the impact parameter and photon energy. Comparisons of the measured cross-sections to QED predictions from the STARlight and SuperChic models are also presented. Tau-pair production measurements can constrain the tau lepton anomalous magnetic dipole moment, and a recent ATLAS measurement using muonic decays of tau leptons in association with electrons and tracks provides one of the most stringent limits available to date.