In this paper, we address rare-event simulation for heavy-tailed Lévy processes with infinite activities. The presence of infinite activities poses a critical challenge, making it impractical to simulate or store the precise sample path of the Lévy process. We present a rare-event simulation algorithm that incorporates an importance sampling strategy based on heavy-tailed large deviations, the stick-breaking approximation for the extrema of Lévy processes, the Asmussen-Rosiński approximation, and the randomized debiasing technique. By establishing a novel characterization for the Lipschitz continuity of the law of Lévy processes, we show that the proposed algorithm is unbiased and strongly efficient under mild conditions, and hence applicable to a broad class of Lévy processes. In numerical experiments, our algorithm demonstrates significant improvements in efficiency compared to the crude Monte-Carlo approach.