Intumescent flame retardants (IFR) are an excellent solution to the problem of easy combustion of polymers. Still, the negative effect of the addition of flame retardants is the decline of the mechanical properties of polymers. In this context, carbon nanotubes (CNTs) are modified with tannic acid (TA) and then wrapped on the surface of ammonium polyphosphate (APP) to construct a special intumescent flame retardant structure (CTAPP). The respective advantages of the three components in the structure are explained in detail, especially the role of CNTs with high thermal conductivity in the flame retardant system. Compared with pure natural rubber (NR), the peak heat release rate (PHRR), total heat release (THR), and total smoke production (TSP) of the composites proposed with special structural flame retardants are decreased by 68.4%, 64.3%, and 49.3%, respectively, while the limiting oxygen index (LOI) increased to 28.6%. The TA-modified CNTs wrapped on the surface of APP can effectively reduce the mechanical damage caused by the flame retardant to the polymer. To sum up, the flame retardant structure of TA-modified CNTs wrapped on APP can effectively enhance the flame retardant properties of the NR matrix and reduce the negative impact on mechanics caused by adding APP flame retardant.