This paper introduces conformal Lyapunov optimization (CLO), a novel resource allocation framework for networked systems that optimizes average long-term objectives, while satisfying deterministic long-term reliability constraints. Unlike traditional Lyapunov optimization (LO), which addresses resource allocation tasks under average long-term constraints, CLO provides formal worst-case deterministic reliability guarantees. This is achieved by integrating the standard LO optimization framework with online conformal risk control (O-CRC), an adaptive update mechanism controlling long-term risks. The effectiveness of CLO is verified via experiments for hierarchal edge inference targeting image segmentation tasks in a networked computing architecture. Specifically, simulation results confirm that CLO can control reliability constraints, measured via the false negative rate of all the segmentation decisions made in the network, while at the same time minimizing the weighted sum of energy consumption and precision loss, with the latter accounting for the rate of false positives.