The anime and manga have quite different storylines. Therefore, whether the prime minister dies in the anime or not is up for debate, but in the manga chapter 227, it explicitly shows the prime minister burning to ashes after overusing the seven flames, as well as a news broadcast in chapter 228 about the death of the prime minister, with the next panel having normal citizens commenting on how his death will affect politics.
It is up to you to use the events of the manga to make sure whether the prime minister dies at the end of the anime. In my opinion, though, I think it's best to keep the events of the anime and manga separate.
One thing to note about my answer: I haven't watched the anime fully yet. I will check the last episode, however, and update this answer accordingly.
Edit: Just saw the ending of the anime. The story of the anime is so different from that of the manga that I think whether the prime minister dies in the anime or not will always just be speculation.