We present new Far Ultraviolet Spectroscopic Explorer (FUSE) spectroscopy of the dwarf starburst galaxy NGC 625. These observations probe multiple phases of the interstellar medium (ISM), including the coronal, ionized, neutral, and molecular gas. This nearby (D = 3.9 ± 0.2 Mpc) system shows a clear detection of outflowing coronal gas as traced by O VI λ1032 absorption. The centroid of the O VI profile is blueshifted with respect to the galaxy systemic velocity by ~30 km s^-1, suggesting a low-velocity outflow. The implied O VI velocity extent is found to be 100 ± 20 km s^-1, which is fully consistent with the detected H I outflow velocity found in radio synthesis observations. We detect multiple lines of diffuse H₂ absorption from the ISM of NGC 625; this is one of only a few extragalactic systems with FUSE detections of H₂ lines in the Lyman and Werner bands. We find a potential abundance offset between the neutral and nebular gas that exceeds the errors on the derived column densities. Since such an offset has been found in multiple dwarf galaxies, we discuss the implications of a lower-metallicity halo surrounding the central star-forming regions of dwarf galaxies. The apparent offset may be due to saturation of the observed O I line, and higher signal-to-noise ratio (S/N) observations are required to resolve this issue.