This paper investigates the information-theoretic energy efficiency of intelligent reflecting surface (IRS)-aided wireless communication systems, taking into account the statistical quality-of-service (QoS) constraints on delay violation probabilities. Specifically, effective capacity is adopted to capture the maximum constant arrival rate that can be supported by a time-varying service process while fulfilling these statistical QoS requirements. We derive the minimum bit energy required for the IRS-aided wireless communication system under QoS constraints and analyze the spectral efficiency and energy efficiency tradeoff at low but nonzero signal-to-noise ratio (SNR) levels by also characterizing the wideband slope values. Our analysis demonstrates that the energy efficiency for the considered system under statistical QoS constraints can approach that for a system without QoS limitations in the low-SNR regime. Additionally, deploying a sufficiently large number of practical IRS reflecting elements can substantially reduce energy consumption required to achieve desired spectral efficiency performance in the low-power regime, even with limited bit-resolution phase shifters. Besides, we reveal that compared with the results applied to the low-power regime, higher effective capacity performance can be achieved in scenarios with sparse multipath fading while achieving the same minimum bit energy in the wideband regime.