Three-dimensional potential energy functions (PEFs) and rovibronic spectra for several low-lying electronically excited singlet states of HCO+ have been investigated based on the ab initio complete active space self-consistent field (CASSCF) method. The calculated electronic transition moments show that the ultraviolet (UV) absorption spectrum of the ion should be dominated by the transition C 1PI-X 1SIGMA+. Analytical representations of the PEFs for the two components of the degenerated C 1PI state have been used in beyond Born-Oppenheimer calculations of the rovibronic energy levels of HCO+ and DCO+ by a Renner-Teller variational approach which accounts for anharmonicity, rotation-vibration, and electronic angular momentum coupling effects. Since the excited states of HCO+ are experimentally unknown up to now, the calculated spectroscopic constants, the vertical and adiabatic excitation energies, and the absorption spectra should provide valuable information for future experimental characterizations of the excited states.