Does heat from gravitational pressure dissipate eventually? It's my understanding that Earth's core is hot and molten because of the high pressure from gravity compressing the planet's mass towards the center.  How does this heat reconcile with thermodynamics, meaning, would it be possible for all the heat to bleed away into space through infrared radiation etc., such that the Earth eventually had about the same mass in about the same configuration, but was instead cold at the core?  Or would it always be the case that as long as the planet holds together, it will be hot at the center due to gravity?
How are these two processes reconciled, meaning, why would there eventually be a "heat death" of the universe so long as there is gravity making masses come together into a form that causes local hot spots (or in larger cases, new suns)?  Wouldn't the planet have to fly apart to get much colder, and why would it do that?  Is it possible for the mass to stay together while all the energy dissipates (ie temperature eventually goes to zero K)?
[I'm not a physicist, so if I'm way off the rails, please help me understand what's really going on.]
 A: Addressing only the question of "Will the Earth grow cold?", the fate of the universe turns on cosmological considerations.
Short answer: Yes.
Slightly longer answer: By measuring the thermal gradient of the crust in deep caves and boreholes at many places on the continental and oceanic crust it is possible to find an approximation to total geothermal power. 
Aside:
A substantial portion of the geothermal flux in this era is from radioactive decay rather than from gravitational potential.
Results published (the link is the the arXiv preprint, but the paper also appeared in Phys. Rev. B) by the Borexino collaboration in 2010 and by the KamLAND collaboration in 2011 (Nature Geoscience) are consistent with roughly half of the geothermal power of the Earth being due to radiological decay. 
These measurements also put strict upper limits on the power of a theorized natural nuclear reactor at the core (the data are now consistent with zero reactor power).
Both Borexino and KamLAND are large anti-neutrino detectors and are directly sensitive to the anti-neutrino emissions of radioactive beta decays such as those found in the Uranium and Thorium chains (but not to Potasium-40 on account of using inverse beta decay as the detection mechanism). From this data we can reconstruct the overall radioactive decay activity in the deep Earth, and compute the total power represented.

Disclaimer: I worked on KamLAND for 3 years, but am not named as an author on  the paper cited herein.
Further disclaimer: much of the text here is adapted from my earlier answer on Skeptics.SE.
A: The earth's core is hot not just because of the gravitational energy released when the earth was formed, but also because of the radioactive elements that contribute energy continuously (radiation from the sun adds a comparatively minor amount). However, eventually all radioactive elements will decay into stable ones and will release no further energy, so the earth will then gradually cool down as it radiates its heat into space. 
This will take many billions of years though, and before then the sun will have become a red giant. This will happen in about 5 billion years. In the red giant phase the sun will expand enough to engulf the earth and destroy it. Then, many billions of years later, the sun too will cool down as it runs out of fuel.
The "heat death" of the universe will eventually happen, but much, much further into the future. It can only happen long after all stars have cooled down - and dwarf stars can take trillions of years to do so. If current guesses are correct, protons will start to decay in about 10^35 or so years. The next step is for black holes to evaporate, something that was predicted by Stephen Hawking. Itd will take 10^67 years or thereabouts for a black hole with the mass of the sun. Giant black holes will take much longer again. After all that, the universe will be filled with with an extremely dilute soup of energy and particles (e.g. 1 electron in a volume the size of the current universe), and nothing else can happen. The universe will finally have come to a halt. The term "heat death" means that the universe has reached the state of maximum entropy and no energy is left for any useful work.
In their book "The Five Ages of the Universe", Adams & Laughlin spend a lot of time talking about this extremely remote future. But remember that it is based on our current ideas. This may well change as we learn more about the universe and about quantum physics. So what will happen in 10^100 or more years is anyone's guess really.
