We examine the star formation history and stellar feedback effects of dwarf galaxies under the influence of extragalactic ultraviolet radiation. Previous work has indicated that the background UV flux can easily ionize the gas within typical dwarf galaxies, delaying or even preventing cooling and star formation within them. Many dwarf galaxies within the Local Group are, however, observed to contain multiple generations of stars, the oldest of which formed in the early epochs of cosmic evolution, when the background UV flux was intense. In order to address this paradox, we consider the dynamical evolution of gas in dwarf galaxies using a one-dimensional, spherically symmetric, Lagrangian numerical scheme to compute the effects of radiative transfer and photoionization. We include a physically motivated star formation recipe and consider the effects of feedback. This scheme allows us to follow the history of the gas and of star formation within dwarf galaxies, as influenced by both external and internal UV radiation. Our results indicate that star formation in the severe environment of dwarf galaxies is a difficult and inefficient process. In potentials with total mass less than a few times 106 M☉ and velocity dispersion less than a few kilometers per second, residual gas is efficiently photoionized by cosmic background UV radiation. Since the density scale height of the gas within these galaxies is comparable to their size, gas may be tidally removed from them, leaving behind starless residual dark matter clumps. For intermediate-mass systems, such as the dSphs around the Galaxy, star formation can proceed within early cosmic epochs despite the intense background UV flux. Triggering processes such as merger events, collisions, and tidal disturbance can lead to density enhancements, reducing the recombination timescale, allowing gas to cool and star formation to proceed. However, the star formation and gas retention efficiency may vary widely in galaxies with similar dark matter potentials because they depend on many factors, such as the baryonic fraction, external perturbation, initial mass function, and background UV intensity. We suggest that the presence of very old stars in these dwarf galaxies indicates that their initial baryonic-to-dark matter content was comparable to the cosmic value. This constraint suggests that the initial density fluctuation of baryonic matter may be correlated with that of the dark matter. For the more massive dwarf elliptical galaxies, the star formation efficiency and gas retention rate are much higher. Their mass-to-light ratio is regulated by star formation feedback and is expected to be nearly independent of their absolute luminosity. The results of our theoretical models reproduce the observed (M/L)-Mv correlation.