The paradigm of the upcoming 5G and beyond is the machine type communications (MTC), where a large number of devices automatically operate wireless communications. Due to their automatic and energy-intensive operations, energy efficiency (EE) becomes very important, but there is a lack of investigation for EE of random access networks, which is the underlying platform for MTC. In this paper, we focus on the EE of random access orthogonal and non-orthogonal multiple access (OMA and NOMA) networks. We first construct mathematical models of their EE performances. Information-theoretic theories are employed to transform probabilistic decodability into deterministic form for OMA, and along with the unique features of NOMA decoding process, by pivoting around the decodability of a specific packet, we realize the modeling for NOMA. Then, based on the established models, exploiting the nature of random access networks, a complementary geometric programming approach is used for performance optimization. From our investigation, the following results are discovered: (1) NOMA has significant advantage over OMA in EE performance due to its strong data recovery; (2) to maximize the EE performance, devices with higher data rate should transmit more frequently, but with lower transmission power; (3) lower data rates have beneficial effects on EE.