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ABSTRACT The Zwicky Transient Facility (ZTF) performs a systematic neutrino follow-up programme, searching for optical counterparts to high-energy neutrinos with dedicated Target-of-Opportunity (ToO) observations. Since first light in March 2018, ZTF has taken prompt observations for 24 high-quality neutrino alerts from the IceCube Neutrino Observatory, with a median latency of 12.2 h from initial neutrino detection. From two of these campaigns, we have already reported tidal disruption event (TDE) AT 2019dsg and likely TDE AT 2019fdr as probable counterparts, suggesting that TDEs contribute >7.8 per cent of the astrophysical neutrino flux. We here present the full results of our programme through to December 2021. No additional candidate neutrino sources were identified by our programme, allowing us to place the first constraints on the underlying optical luminosity function of astrophysical neutrino sources. Transients with optical absolutes magnitudes brighter that −21 can contribute no more than 87 per cent of the total, while transients brighter than −22 can contribute no more than 58 per cent of the total, neglecting the effect of extinction and assuming they follow the star formation rate. These are the first observational constraints on the neutrino emission of bright populations such as superluminous supernovae. None of the neutrinos were coincident with bright optical AGN flares comparable to that observed for TXS 0506+056/IC170922A, with such optical blazar flares producing no more than 26 per cent of the total neutrino flux. We highlight the outlook for electromagnetic neutrino follow-up programmes, including the expected potential for the Rubin Observatory.