In this paper, we investigate energy-efficient resource management for downlink OFDMA systems with feedback channel capacity constraints. Compared with previous works, quantized channel state information (CSI) is studied by using rate-distortion theory since it can establish an information-theoretic lower bound on the capacity of the feedback channel. Based on the quantized CSI, an effective resource allocation algorithm is proposed to maximize the system energy efficiency employing the generalized fractional programming theory and the Lagrange dual decomposition method. Numerical results show that our proposed algorithm can nearly achieve the optimal solution and imply that the energy efficiency with a limited feedback rate could be close to that with perfect CSI.