In this paper, we introduce an analytical approach for modeling and analyzing the performance of multi-tier cellular networks that are powered by the power grid and by renewable energy sources. The proposed approach relies on modeling the locations of the base stations, either powered by the power grid or by renewable energy sources, by using Poisson point processes. The availability of renewable energy is modeled by using a Poisson point process in the time domain. In addition, the temporal dynamics of the batteries of the base stations powered by renewable energy sources are modeled by using a discrete Markov chain with a number of states that is equal to the finite storage capacity of the batteries. In particular, the coverage probability, the spectral efficiency, and the energy efficiency of the considered network model are formulated in an analytical, closed-form, expression, which depends on the probability that the typical base station is available, i.e., it has sufficient power to serve at least one mobile terminal in its cell. This latter probability is shown to be the solution of the steady state equation of the Markov chain that models the temporal dynamics of the batteries of the base stations. Under the assumption that the batteries of the base stations can be either empty or fully charged, we formulate an optimization problem in order to maximize the energy efficiency as a function of the transmit power and the deployment density of the base stations, and identify sufficient conditions for which the problem admits a unique solution. The accuracy of the proposed approach and the performance trends inferred from it are substantiated with the aid of extensive Monte Carlo simulations.