Is it possible that nuclear fission contributes to climate change? This is probably a really stupid question, please forgive me.
Is it possible that the use of nuclear fission on earth contributes to the increased energy in the Earth's system as according with the Laws of Thermodynamics regarding the Conservation of Energy?
 A: No, it is not possible that nuclear fission is responsible for climate change.  So although it's theoretically possible that we could do enough nuclear fission to raise the global heat content of the earth significantly, in reality, we've done very little nuclear fission industrially, and it's a tiny tiny contribution.
First, remember that burning fossil fuels also adds to the global heat content. And that way way outstrips nuclear fission: each year, nuclear contributes to about 6% of global energy demand. That's less than we get from renewables, and much much less than we get from natural gas, coal and oil (those last three total around 80% of the global energy supply). So the contribution to the global heat content from nuclear is smaller than that from fossil fuels.  All in all, the combined output from nuclear, oil, gas and coal is the equivalent of about 12TW of extra heat into the Earth (in mean power terms).
Remember, as far as global heat content is concerned, releasing locked-up chemical energy is no different to releasing locked-up nuclear energy.  They're both just the introduction of extra energy into global heat content; energy that would otherwise have been locked away in non-heat form: either as mass, or as chemical energy.
And the 12TW of extra heat is tiny compared to the impact on global heat content from anthropogenic greenhouse gases.
The Earth's surface is around $5\times 10^{14} m^2$, so 12TW is about $0.025W / m^2$. The radiative forcing from $CO_2$ is around $1.8W / m^2$
Increasing the atmospheric concentrations of greenhouse gases directly (in particular, $CO_2$, but also $CH_4$ and several others), and indirectly, through secondary feedbacks and forcings, causes an increase in global heat content that is orders of magnitude larger than the heat added to the Earth from all human activities.
Thanks to AlanSE for noting that in general, renewables don't contribute to global heat content: broadly speaking, the energy in hydro, wind, wave and solar, would all have ended up as low-grade heat anyway. Geothermal was obviously part of the global heat content to start with. One might argue that tidal power does add to global heat content, but that's absolutely miniscule amounts. Biomass might in some cases have added to global heat content; in other cases, it might add something to embodied energy locked up in soil chemistry.  There's also the effect of changes in global albedo from changes in land-use for energy: but again, these are miniscule in comparison with the forcing effect of greenhouse gases.
A: Other answers covered that the heat added is negligible. The only other thing to consider is greenhouse gas emitting process involved with making the fission possible. Mining, refining, transporting of the fuel all likely involves lots of fossil fuels. 
