In this paper, we consider the complexity of the minimum feedback vertex set problem (MFBVS) for tournaments with forbidden subtournaments. The MFBVS problem in general tournaments is known to be NP-complete. We prove that the MFBVS problem for $W_5$-free and $U_5$-free tournaments is in P, and for $T_5$-free tournaments it remains NP-complete. Moreover, we prove a necessary condition for all $H$ such that the MFBVS problem for $H$-free tournaments is in P. We also show that the necessary condition is not sufficient.