The effect of nonmagnetic impurities on the superconducting transition temperature T_c in superconductors with a non-phonon-mediated (nonphononic) mechanism of superconductivity is investigated. Two features inherent to high-T_c superconductors are taken into account: the overlap of the energy bands, and the variable density of charge carriers. The cases of s and d_x2-y2 symmetry of the order parameter are considered. It is demonstrated that the Anderson theorem is violated in the case of s symmetry of the order parameters, both on account of interband scattering of electrons on impurities and on account of intraband scattering due to the electron-hole asymmetry in systems with a nonphononic mechanism of superconductivity. Analytical expressions are obtained for the value of T _c in the regions of small and large impurity concentrations, and numerical solutions of a self-consistent system of equations are found. The influences on T_c due to various mechanisms (the change in the chemical potential and in the intraband and interband relaxation times at an impurity) are analyzed for the types of symmetry indicated above. It is shown that a number of the features inherent to high-T_c superconductors can be explained, including, in particular, the possibility of complete suppression of superconductivity by a nonmagnetic impurity and the possibility of attaining high values of T_c upon doping of an insulator. It is also shown that as the interband scattering on the impurity potential increases, there is a change in the dependence of T_c on the impurity concentration. Qualitative agreement is obtained between the proposed theory and the experimental data on the dependence of T_c on the concentration of Zn, Al, and Ga impurities in yttrium and lanthanum ceramics.