Could the Earth's core lose its heat?
Part 1, see Neos answer.  Earth will lose its heat no matter what we do, and our extraction of geothermal energy is insignificant (Wikipedia quotes a BP figure of 11.4 GW electrical, 28 GW heating).

To answer part 2 of your question: if the Earth's core loses its heat, this will not have a major direct impact on climate.  Internal heat generation is estimated by Davies and Davies (2010) to be roughly 47 TW.  With a surface area of 5.1 &times; 1014 m2, this translates to roughly 0.1 W/m2.  This can be compared to the other flows in the climate system, illustrated by Trenberth and Fasullo, 2012:


&mdash;Trenberth, Kevin E., and John T. Fasullo. "Tracking Earth’s energy: From El Niño to global warming." Surveys in geophysics 33, no. 3-4 (2012): 413-426.  Weblink

So, from a climate perspective, internal heat generation is not important.  See also this post on skepticalscience.

However, we might lose our atmosphere, which would have inconvenient consequences.  An ice age would be the least of our worries.  A subsequent question would be: (How long) would Earth&#39;s atmosphere last without a global magnetic field?  That is a different question and I'm not sure if we really know the answer.