The inner core of the Earth has an enormous temperature of around 4300 degrees Celsius. Is this large inner core temperature an important factor contributing to the ideal temperature that is necessary to maintain life as we know it on the Earth's outer-crust and atmosphere, or is the temperature of the core quite irrelevant to the temperature on the Earth's crust?
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 °C, and even a more realistic estimate of the surface temperature without an atmosphere is around 1 °C . 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 . 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.
 Martin, W., J. Baross, D. Kelley, M.J. Russell (2008) Hydrothermal vents and the origin of life. Nature
Yes and no. The Earth's hot inner core is one of many factors that contribute to the theory of Rare Earth indirectly. Since the potential temperature difference is what drives mantle convection (well density, but density depends on both pressure and temperature), the cold lithosphere and the hot core are absolutely necessary for the development of complex life if you take Darwin's theory of evolution to be the primary driver of speciation. Though, I'm sure there are other environments where life can emerge (I'm hoping one day we will find methanogenic life on Titan, that would be cool).
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.
The liquid outer core and the magnetic field it generates are essential to life. Without the magnetic field we would have no atmosphere. The temperatures would be roughly equal to the moon (Which goes from 123 C on the hemisphere that happens to face the sun and -153 C on the shadowed hemisphere).
Source for moon temperatures:
The earth would be dead without a molten core. There would be no volcanoes, earthquakes etc no plate tectonics. Any material eroded from mountains would eventually end up at the bottom of the sea, and then stay there. Eventually there would be no land above sea level. Erosion would continue until we are left with nothing but deep seas with little or no mineral nutrients. Not great for a rich diversity of life.
So when we hear of the natural disasters of volcanoes and earthquakes, appreciate that the very earth is alive and sustaining us.
The theory of a Gravity Induced Lapse Rate proposed by Loschmidt (1876) suggests that all planetary bodies will have a temperature gradient in their atmosphere AND in the solid structure of the planet all the way to the core. If this theory is correct then the hot inner core of planets with an atmosphere is a common feature of all such bodies. It is (we think) obviously the case that life does not form on all such bodies but only on a select few. The features of the hot core, cooler surface & gaseous atmosphere would not then be the deciding factor as to whether we have a lifeless planet or not. If we assume, like the question, that life requires a certain temperature range then the main deciding factor will be the surface temperature as we expect life to be moving, swimming or flying about pretty close to the surface (generally). The 2 main deciding factors would be distance from the star and composition of the atmosphere, next factor is probably rotational speed on axis.
A molten core is essential for the survival of life on Earth as we know it. The molten core of our planet is responsible for generating a magnetic field that shields the surface from harmful solar radiation and cosmic rays. Without this protective magnetic field, the Earth's atmosphere would be stripped away, and life would likely not be able to survive.
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.