First-principles calculation of the critical temperature of a superconductor Special materials become (conventional) superconductors at a specific temperature, referred to as the critical temperature.
Are there any techniques for calculating from first principles whether a material is such a material, and if so, the critical temperature? If not, what is the obstacle?
 A: TLDR; No there aren't. Many body physics is not easy to solve.
Convential Superconductivity:
The reason why it is not possible is that convential superconductivity, as we understand it today, involves quantum mechanical states that are extended over a large number of lattice sites (it is a phenomenon involving long range order - mesoscopic physics). The cooper pairs involved can extend over hundreds of lattice sites. Therefore we would need to take at the least a few hundred thousand of such lattice sites into consideration if we want reliable calculation results for conventional superconductivity.
Let's look at the analytical and numerical methods with which the problem of superconductivity can be addressed and why they are necessarily unable to describe high $T_c$ superconductivity.
Solving the first principles Hamiltonian analytically:
Not possible today! We can't even solve the three body problem in classical mechanics. No chance to solve a thousand body problem in quantum mechanics. What we can do is write down an approximate Hamiltonian which does a good job at describing the experimental findings. This gives you the BCS theory.
What about numerical methods
The basic challenge of numerical many body methods is the amount of storage needed for the calculations. Lets consider the simplest of examples a spin 1/2 system consisting of $N$ particles. To describe a state in this system requires $2^N$ Bits. A system consisting of 64 particles would require roughly 2 Million Terrabytes of storage.
Now lets look at some details:
If you think of first-principles calculations as calculations originating from what we know about atoms, their constituents and their surrounding electrons: No there aren't any such calculations. Atomic physics is already struggling to accurately describe the energy levels of molecules without approximations from first principles. With increasing numbers of electrons involved in the molecule one will eventually have to resort to approximations, or otherwise the many body problem becomes intractable.
If you think of first-principles calculations as calculations originating from what we know about how atoms interact (usually only taking into account electron-electron interactions), no there aren't such calculations. The most one can do realistically today is probably around 10 particles, no more. One group has done calculations with electron-electron (http://journals.aps.org/prb/abstract/10.1103/PhysRevB.80.045326) and contact interactions (http://iopscience.iop.org/article/10.1088/1367-2630/18/7/073018/meta) in the past (unrelated to super conductivity) and the computational effort necessary to achieve convergence for those calculations is huge. The reason is that you need to store the wavefunctions and not just their spins! This makes the memory requirements even worse!
Long story short, it is very unlikely that we will be able to solve conventional superconductivity from first principles with our classical computers. And neither does it look like we will able to do so with analytical methods. We have to resort to approximations and effective Hamiltonians.
Unconventional Superconductivity:
Here our chances might be a little better. Why? Because it is stipulated by part of the community that unconventional superconductivity might be a local phenomenon (at least regarding the cuprates). So we might only need a few dozen lattice sites in each direction to fully capture the physics. Now 12^3=1728 is still much more than what we would ever be able to solve from first principles, but at least there is some hope that finite size calculations might give at least a hint of the right answer. This is a field of active research using approximate Hamiltonians like the Hubbard model.
So what can we do? Is all hope lost?:
No! Effective (approximate) Hamiltonians can do a terrific job at describing physics. BCS theory has been doing very well, the Hubbard Model too has proven itself for very small systems (http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.114.080402).
One usually doesn't need all the microscopic details to accurately describe many body physics. That is why so much effort is put into approximate Hamiltonians and into figuring out which one of the theories is the correct one.
An interesting approach is to simulate quantum systems with another quantum system (an idea originated by Richard Feynman, https://doi.org/10.1007/BF02650179). If we have a quantum system that we have ultimate control over we could set it up to reflect the properties of a superconducting material. We could then use it to study the super conducting effects. Does superconductivity vanish when I change the interaction strength? What lattice parameters give the highest $T_c$, etc. There are many experimental efforts directed towards that goal (http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.114.080402 , http://link.aps.org/doi/10.1103/PhysRevLett.108.205301 , http://www.sciencemag.org/cgi/doi/10.1126/science.aag1430 and many more)
A: It is hard to give a completely general answer (and I am not an expert), but I think for conventional (weak coupling electron-phonon) superconductors the answer is yes. There is certainly a well defined procedure: 
1) Determine the electronic states near the Fermi surface (using highly improved DFT functionals, for example). 
2) Determine the phonon spectrum. If you have an energy functional that predicts the correct crystalline structure this can be reliably done. 
3) Determine the electron-phonon coupling. This is harder than the first two, but if you know the phonon modes, an electronic DFT should provide you with predictions of the electron-phonon coupling.
4) Use Eliashberg-theory (a version of BCS that involves the full phonon propagator, screened Coulomb forces, and the phonon correction to the electron self energy) to predict Tc. The main difficulty is that Tc is exponentially sensitive to the electron-phonon coupling, so that even small errors in the coupling can lead to sizable uncertainties in Tc.
See, for example, https://arxiv.org/abs/1601.03486 for a recent example with some theory-experiment comparisons. 
