This paper proposes optimal power adaptation schemes that maximize the energy efficiency (EE) in the presence of Markovian sources and finite-alphabet inputs subject to quality of service (QoS) constraints. First, maximum average arrival rates supported by transmitting signals with arbitrary input distributions are characterized in closed-form by employing the effective bandwidth of time-varying sources (e.g., discrete-time Markov and Markov fluid sources) and effective capacity of the time-varying wireless channel. Subsequently, EE is defined as the ratio of the maximum average arrival rate to the total power consumption, in which circuit power is also taken into account. Following these characterizations, an optimization problem is formulated to maximize the EE of the system, and optimal power control schemes are determined. Through numerical results, the performance of the optimal power control policies is evaluated for different signal constellations and is also compared with that of constant power transmission. The impact of QoS constraints, source characteristics, input distributions on the maximum achievable EE and the throughput is analyzed.