The properties of interstitial and substitutional Li in wurtzite ZnO are modeled using hybrid density functional calculations. We investigate the impact of the band-gap error on the formation energies of the two defects and their dependence on the Fermi level. It is found that within a local-density approximation, the acceptor level of LiZn is very close to the valence-band top but as the band gap is opened, the acceptor state becomes more localized and the respective level is shifted upward. Taking polaronic effects into account, we place the ionization level of LiZn between Ev+0.60 eV and Ev+1.1 eV. This deeper level explains the difficulty in realizing p-type ZnO using Li as monodopant. Further, the mobility of the defects was investigated. While interstitial Li is mobile at low temperatures, independent of the stoichiometry, the diffusion of LiZn depends on the concentrations of intrinsic defects. Our calculations show that in O-rich material, where the defect is more stable, the dominant diffusion process corresponds to a dissociative mechanism requiring a substantial activation energy.