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This paper studies online resource allocation with replenishable budgets, where budgets can be replenished on top of the initial budget and an agent sequentially chooses online allocation decisions without violating the available budget constraint at each round. We propose a novel online algorithm, called OACP (Opportunistic Allocation with Conservative Pricing), that conservatively adjusts dual variables while opportunistically utilizing available resources. OACP achieves a bounded asymptotic competitive ratio in adversarial settings as the number of decision rounds T gets large. Importantly, the asymptotic competitive ratio of OACP is optimal in the absence of additional assumptions on budget replenishment. To further improve the competitive ratio, we make a mild assumption that there is budget replenishment every T* ≥ 1 decision rounds and propose OACP+ to dynamically adjust the total budget assignment for online allocation. Next, we move beyond the worst-case and propose LA-OACP (Learning-Augmented OACP/OACP+), a novel learning-augmented algorithm for online allocation with replenishable budgets. We prove that LA-OACP can improve the average utility compared to OACP/OACP+ when the ML predictor is properly trained, while still offering worst-case utility guarantees when the ML predictions are arbitrarily wrong. Finally, we run simulation studies of sustainable AI inference powered by renewables, validating our analysis and demonstrating the empirical benefits of LA-OACP.