Municipal solid waste incineration fly ash (MSWI FA) is commonly managed on the basis of short-term leaching compliance, yet its long-term behavior under landfill-relevant conditions remains insufficiently resolved. This review summarizes published studies on heavy-metal occurrence, leaching controls, stabilization/solidification performance, and landfill aging processes in MSWI FA. Particular attention is given to the compositional heterogeneity of FA across waste sources and incineration systems, the metal-specific and pH-dependent nature of leaching behavior, and the long-term stability limitations of current treatment strategies. The available evidence indicates that heavy-metal release is governed not only by initial immobilization efficiency, but also by evolving geochemical and physical conditions during landfill aging, including carbonation, redox shifts, chloride interactions, wet-dry cycling, and freeze-thaw damage. Field and simulation studies further show that materials meeting short-term regulatory criteria may still exhibit substantial re-leaching risks over time, especially in cement-based stabilization systems exposed to complex disposal environments. These findings indicate that MSWI FA management should account for ash composition, treatment history, local hydroclimatic conditions, and expected landfill exposure. This review summarizes the evidence needed to evaluate long-term stability and to support scenario-specific management of MSWI FA under landfill conditions.