Thermal photons from the photosphere may be the primary source of the observed prompt emission of gamma-ray bursts (GRBs). In order to produce the observed non-thermal spectra, some kind of dissipation mechanism near the photosphere is required. In this paper we numerically simulate the evolution of the photon spectrum in a relativistically expanding shell with a time-dependent numerical code. We consider two basic models. One is a leptonic model, where a dissipation mechanism heats the thermal electrons maintaining their high temperature. The other model involves a cascade process induced by pp(pn)-collisions which produce high-energy electrons, modify the thermal spectrum, and emit neutrinos. The qualitative properties of the photon spectra are mainly determined by the optical depth at which the dissipation mechanism sets in. Too large optical depths lead to a broad and curved spectrum contradicting the observations, while for optical depths smaller than unity the spectral hardness becomes softer than observed. A significant shift of the spectral peak energy to higher energies due to a large energy injection can lead to an overly broad spectral shape. We show ideal parameter ranges for which these models are able to reproduce the observed spectra. For the pn-collision model, the neutrino fluence in the 10-100 GeV range is well above the atmospheric neutrino fluence, but its detection is challenging for presently available detectors.