Passive red galaxies frequently contain warm ionized gas and have spectra similar to low-ionization nuclear emission-line regions (LINERs). Here we investigate the nature of the ionizing sources powering this emission, by comparing nuclear spectroscopy from the Palomar survey with larger aperture data from the Sloan Digital Sky Survey. We find the line emission in the majority of passive red galaxies is spatially extended; the Halpha surface brightness profile depends on radius (r) as r^(-1.28). We detect strong line ratio gradients with radius in [N II]/Ha, [S II]/Ha, and [O III]/[S II], requiring the ionization parameter to increase outwards. Combined with a realistic gas density profile, this outward increasing ionization parameter convincingly rules out AGN as the dominant ionizing source, and strongly favors distributed ionizing sources. Sources that follow the stellar density profile can additionally reproduce the observed luminosity-dependence of the line ratio gradient. Post-AGB stars provide a natural ionization source candidate, though they have an ionization parameter deficit. Velocity width differences among different emission lines disfavor shocks as the dominant ionization mechanism, and suggest that the interstellar medium in these galaxies contains multiple components. We conclude that the line emission in most LINER-like galaxies found in large aperture (>100pc) spectroscopy is not primarily powered by AGN activity and thus does not trace the AGN bolometric luminosity. However, they can be used to trace warm gas in these red galaxies. ; Comment: 27 pages, 25 figures, ApJ in press; v2: added Figure 20 and made minor revisions matching the accepted version