What is the symmetry which is responsible for conservation of mass? According to Noether's theorem, all conservation laws originate from invariance of a system to shifts in a certain space. For example conservation of energy stems from invariance to time translation.
What kind of symmetry creates the conservation of mass?
 A: Noether's theorem says that symmetries lead to conservation laws, not the converse. Conservation of mass doesn't follow from any of the obvious symmetries of nonrelativistic motion. Those symmetries are translations in space (leading to conservation of momentum), translations in time (conservation of energy), rotations (conservation of angular momentum), and boosts (i.e. changes to a frame moving at constant velocity with respect to the original frame, leading to conservation of center-of-mass motion). The algebra of these symmetries, known as Galilean transformations, involves mass as a central charge. Since it commutes with everything in sight, I think it's fair to say that there's no nontrivial associated symmetry.
A: Mass is only conserved in the low-energy limit of relativistic systems.  In relativistic systems, mass can be converted into energy, and you can have processes like massive electron-positron pairs annhillating to form massless photons.  
What is conserved (in theories obeying special relativity, at least) is mass energy--this conservation is enforced by the time and space translation invariance of the theory.  Since the amount of energy in the mass dominates the amount of energy in kinetic energy ($mc^{2}$ means a lot of energy is stored even in a small mass) for nonrelativistic motion, you get a very good approximation of mass conservation. out of the energy conservation.  
A: Noether's relationship implies that for every conserved quantity, there is a symmetry, and vice versa.
In the real world, mass may be converted to energy. For example, uranium power plants convert about 0.1% of the uranium mass to a huge energy, according to the $E=mc^2$ formula. So up to the conventional factor $c^2$, the total energy - including the latent one - and the total mass is the same thing.
Its conservation is linked to the time-translational symmetry of the laws of physics.
In the old world "before Marie Curie", people didn't know any relativity or any other indications of relativistic physics such as radioactivity. So they believed that mass was conserved even if the energy is not included. In their understanding of the world, the total mass of the Universe was the sum of the rest masses of the electrons, protons, and other massive particles.
The conservation law for the mass defined in this way didn't lead to any symmetries because the definition only depends on parameters - masses of all point masses are parameters - and not on dynamical, time-dependence quantities such as the positions or velocities. That's why Noether's theorem didn't apply.
It is not a real flaw of Noether's theorem because, as we know today, the total mass defined in the old way - and neglecting the mass increases from velocity (kinetic energy) and other forms of energy - is actually not conserved. This is no coincidence; a world compatible with general relativity doesn't allow any mass-like quantities to be non-dynamical. All quantities describing particular objects have to be dynamical i.e. changeable, and that's why Noether's theorem always holds in the world.
A: The quantity that is conserved is the squared momentum four-momentum $p^2$ of the entire universe (or any isolated system), caused by the Poincaré invariance. It also results in the conservation of total spin by the way. Wigner's classification provides some interesting further reading, too.
A: For a superfluid, mass generates a change in the phase. That's because a superfluid is a superposition of states with different masses, i.e. a mass condensate. This also applies to supersolids. For other cases, states with different masses lie in different superselection sectors.
You might object that for say, a helium-4 superfluid, a change in phase generates the number of helium-4 atoms, and not the mass per se, and that's true. But going the other way around, mass generates a change in the phase, scaled by the mass of a helium-4 atom. The reason for this asymmetry is the total number of other forms of matter decomposes the state space into superselection sectors.
This entire analysis presupposes Galilean relativity instead of special relativity. For the latter, mass isn't conserved.
A: It seems that space-time translations leads to an approximate conservation for mass, in the non-relativistic limit. Let's see how:

*

*For a relativistic perfect fluid, we have the following energy-stress tensor:

$$T^{\alpha \beta} \, = \left(\rho + {p \over c^2}\right)u^{\alpha}u^{\beta} + p g^{\alpha \beta}$$


*Noether's theorem for infinitesimal translations takes the form:

$$\frac{\partial T^{\alpha \beta}}{\partial x^\alpha} = 0$$
Substituting this approximations for the components $T^{0\alpha}$:
$$\frac{1}{c}\frac{\partial T^{00}}{\partial t} + \frac{\partial T^{01}}{\partial x} + \frac{\partial T^{02}}{\partial y} + \frac{\partial T^{03}}{\partial z} = 0$$


*For a non-relativistic, pressure-free $(p = 0)$ fluid in Minkowskian spacetime, we have that $u^0 \approx c, u^1 \approx v_x, u^2 \approx v_y$ and $u^3 \approx v_z$, the above equation reduces to:

$$\frac{1}{c}\frac{\partial (\rho c)}{\partial t} + \frac{\partial (\rho v_x)}{\partial x} + \frac{\partial (\rho v_y)}{\partial y} + \frac{\partial (\rho v_z)}{\partial z} = 0$$
This is the continuity equation, which is local from for the conservation of mass in classical mechanics.
A: The rest mass of a particle in Classical Electrodynamics (as well as the particle charge) is a phenomenological parameter (number) in differential equations of motion. It is defined to be constant. No dynamics can change it, just by definition. It is not expressed via dynamical variables and its conservation is not due to some symmetry. $dm/dt = 0, de/dt = 0$ are experimental facts, if you like. Of course, physical models and their equations should be compatible with such definitions. Some "theoretical" hypothesis are incompatible with it, for example, self-acting particles.
If one obtains "corrections" to the particle mass (or/and  charge) in course of perturbative calculations, then the theory formulation is wrong. In some theories they discard such "corrections" and call it "renormalizations". P. Dirac was unhappy about that "renormalization prescription" and invited researchers to change the original equations, not the results.
In Classical Electrodynamics the sum of particle masses $\sum m_i$ is also conserved but it does not define the system mass (the latter is just defined differently).
