The spin pumping in ferromagnet/nonmagnetic materials (FM/NM) heterostructures is an effective technique to inject spin current into NM, which provides the possibility of ultrafast, low-power consumption spintronic devices. Previous work demonstrates that the high order terms in k·p theory can induce a large warping effect, and consequently snowflakelike Fermi contour in topological insulator (TI). Unfortunately, the effect of warped topological surface state (TSS) on pure spin absorption in FM/TI heterostructures is completely unknown. Here, by considering strength of spin accumulation, we identified a mechanism for the anisotropic absorption of spin current in a FM/TI heterostructure with a large warping effect. If the density of states (DOS) of TSS dominates at the Fermi surface, the warping effect results in an anisotropic Gilbert damping at nanosecond timescale and almost isotropic ultrafast demagnetization time at femtosecond timescale. The anisotropy of Gilbert damping is found to decrease as bulk state contributions become more significant in thicker Bi_{2}Te_{3} films. Our theoretical predictions regarding the warping effect-induced spin current absorption at nanosecond and subpicosecond timescales have been experimentally validated in Fe/Bi_{2}Te_{3} heterostructures by ferromagnetic resonance and time-resolved magneto-optical Kerr effect (TRMOKE) results, respectively. Our work provides a more intuitional way to understand the spin transfer mechanism, and lays the groundwork for advancing anisotropic spintronics.