As winwaed also mentioned, the hot inner core is surrounded by a chaotically convecting liquid core which produces the Earth's magnetic field, undoubtedly important to life.
Edit for further explanation on the link between evolution and the potential temperature difference between the core and lithosphere:
While this goes a bit out of my expertise, the Theory of Evolution requires a diversity of environments in which random mutations naturally give a species an advantage over another. Plate tectonics enhances this diversity, by not only transporting land to different climates but also making new ones (arc volcanism). Without the temperature difference between the core and the lithosphere, we would not have mantle convection, which is the fundamental driver of plate tectonics. So while the temperature of the core is not directly necessary for life, it has allowed for a much more diverse speciation on this planet. This, of course, has other implications (such as surviving catastrophic extinction) but those I feel are outside the scope of this question.
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
Source for moon temperatures:
So when we hear of the natural disasters of volcanoes and earthquakes, appreciate that the very earth is alive and sustaining us.
In addition, the molten core plays a critical role in regulating the Earth's climate and maintaining a stable environment for life to thrive. Without the heat generated by the core, the Earth's surface would be much colder and less hospitable to life. The movement of the molten material in the core also helps to drive the motion of the tectonic plates, which plays a critical role in the cycling of nutrients and the regulation of the carbon cycle, which are both important for life on Earth.