Very extended regions of highly ionized gas have been observed in several classes of active galaxies. The distribution of this emitting gas sometimes resembles a cone with apex at the nucleus and extending beyond the classical narrow-line region. There is evidence that photoionization by the active nucleus is the most likely excitation mechanism. The conical shape suggests collimation or anisotropy of the ionizing continuum. On the basis of these facts, we investigate the possibility that photoionization by a geometrically thick, radiation-supported accretion disk can reproduce the emission-line spectra. Such a torus naturally provides the apparent collimation of the ionizing radiation field, because the hot funnel of the torus is occulted when viewing at large angles from the disk axis. We use a simple model for a thick accretion disk around a supermassive black hole and a photoionization code to make theoretical predictions of the line ratios. The sensitivity of the predicted line ratios to variations in ionization parameter, black hole mass, and accretion disk size is discussed in detail. The majority of line ratios are most sensitive to the ionization parameter, but some are also functions of the accretion disk size at fixed ionization parameter. Our results are compared with observational data (for Seyfert galaxies), in order to constrain the disk parameters and the properties of the emission-line clouds. Typical model fits involve an ionization parameter of several times 10-2 and an accretion disk of radius ≃500 (150) gravitational radii around a 106 M⊙ (107 M⊙) black hole. The spectral shape of the ionizing continuum is briefly discussed. The inclusion of additional effects, such as a hard X-ray component and photoelectric absorption of soft X-rays by gas between the nucleus and the extended emission-line regions, may improve the agreement between observed and predicted line ratios.