I understand your thinking: the law of conservation of energy is ironclad. You cannot create or destroy energy, and yet, that's what nuclear reactors (and bombs) seem to be doing.
Well, you're right and wrong. You can't just create energy...but you CAN turn mass into energy. As it turns out, mass is really just a super-compact form of energy. And when I say super-compact, I mean super-duper-ultra-extreme-compact. 1 kilogram of matter (2.2 lbs, on Earth) is equivalent to 9.00x10^16 Joules of energy, or 21.5 Megatons of TNT. That's more than 1400 times the energy released by the bomb that destroyed Hiroshima, Japan, in 1945. And all from one kilogram of matter.
In nuclear reactors, the entire mass of the nuclear fuel is NOT converted to energy. In fact, most of the mass that goes IN to a nuclear reactor also comes back out.
In a reactor, neutrons are used to strike and break apart heavy nuclei like uranium-235. The mass of the U-235 atom and the neutron that splits it is just a tiny bit more than the mass of the two resulting nuclei and any neutrons that they spit out. That loss in mass represents the portion that became energy. Remember, MOST of the particles' mass remains intact in a nuclear change like fission.
So nuclear reactions don't really violate any laws of physics (if they did, we'd have to seriously investigate the laws or conclude that they're not really happening!). Your original question, however, was if the heat produced by nuclear reactions contributes to global warming.
The answer is: not very much. Global warming is caused by more sunlight (particularly infrared, or IR, light) getting trapped close to the Earth's surface by an increase in "greenhouse gases", such as CO2. The vehicles that mine and transport nuclear materials produce CO2, but on the whole, nuclear power is much less carbon-intensive than coal-based power. Also, while some heat escapes the plant and raises the temperature of the plant's surroundings, the amount of heat added to the atmosphere by nuclear power plants pales in comparison to that added by the Sun.
So, again...nuclear reactions don't violate the law of conservation of matter/energy, and I do not believe that nuclear power plants contribute significantly to global warming.
I hope that helps. Good luck!