We present systematic case studies to investigate the sensitivity of axion searches by liquid xenon detectors, using the axio-electric effect (analogue of the photoelectric effect) on xenon atoms. Liquid xenon is widely considered to be one of the best target media for detection of WIMPs (Weakly Interacting Massive Particles which may form the galactic dark matter) using nuclear recoils. Since these detectors also provide an extremely low radioactivity environment for electron recoils, very weakly-interacting low-mass particles (< 100 keV/c^2), such as the hypothetical axion, could be detected as well - in this case using the axio-electric effect. Future ton-scale liquid Xe detectors will be limited in sensitivity only by irreducible neutrino background (pp-chain solar neutrino and the double beta decay of 136Xe) in the mass range between 1 and 100 keV/c^2. Assuming one ton-year of exposure, galactic axions (as non-relativistic dark matter) could be detected if the axio-electric coupling g_Ae is greater than 10^-14 at 1 keV/c^2 (or $10^-13 at 100 keV/c^2). Below a few keV/c^2, and independent of the mass, a solar axion search would be sensitive to a coupling g_Ae ~ 10^-12. This limit will set a stringent upper bound on axion mass for the DFSV and KSVZ models for the mass ranges m_A < 0.1 eV/c^2 and < 10 eV/c^2, respectively. Vector-boson dark matter could also be detected for a coupling constant alpha'/alpha > 10^-33 (for mass 1 keV/c^2) or > 10^-27 (for mass 100 keV/c^2).