Is the Earth's hot inner core necessary for life?
Whether the high temperatures of the core is strictly necessary to maintain a habitable temperature at the surface I'm not sure, as the global temperatures are largely controlled by insolation and the greenhouse effect. The black-body radiation temperature of the Earth is about -18&#8239;&#176;C, and even a more realistic estimate of the surface temperature without an atmosphere is around 1&#8239;&#176;C [1]. That being said, it depends on what you define as life, if the Earth had no atmosphere the temperature would have been lower, but organisms would have adapted to that if they existed in the first place.

As Neo and winwaed also mentioned, the molten iron in the core does protect organisms against harmful radiation from the sun, by providing the Earth with a shielding magnetic field.

Also, the hot core and mantle most likely did influence the origin of life to begin with, as near hydrothermal vents in the ocean floor conditions might have been or are suitable for spontaneous reactions that lead to the formation of self-reproducing molecules [2]. Note this is not in the Darwinian sense Neo talks about (Darwin's book is called On the Origin of Species, not On the Origin of Life), but in the microbiological and geological sense.

[1] https://en.wikipedia.org/wiki/Black-body_radiation#Temperature_of_Earth
[2] Martin, W., J. Baross, D. Kelley, M.J. Russell (2008) Hydrothermal vents and the origin of life. Nature