Does Noether's theorem apply to entropy? Entropy appears to have a translation symmetry - adding some constant value to it doesn't appear to my fairly rudimentary understanding of physics alter the actual physics. Is this correct?
Now (translational) symmetry enters into Noether's theorem; does then this theorem apply to entropy? If it does - what then is the Noether charge?
 A: Noether's theorem states that if a system has a continuous symmetry, there is a quantity related to this symmetry, called the Noether charge, which is conserved.
It does not state anything on the fact that adding a constant term to a measurable quantity may or may not change the physical description of the system. Only some physical quantities in fact are defined up to a constant term (one can add a constant term without change the physics of the system). 
These quantities are, for example, some forms of potential energy, angles and angular phases, but not entropy. 
Entropy is not defined up to a constant term.
Adding a constant term to the entropy does change the physics of the system. For example, the 3rd law of thermodynamics states that the entropy of a perfect crystal (or an ideal gas) at zero temperature is zero. This has the very physical consequence that zero temperature can be reached only asymptotically.

Few clarifications of the Noether's theorem
Examples of continuous symmetries: 
time invariance, translational invariance, rotational invariance. 
Corresponding conserved charge:
energy, linear momentum, angular momentum.
Now, if a system is translational invariant (for example, an isolated and closed system), that means that any thermodynamical observable of the system is translational invariant, e.g., volume, temperature, energy, and entropy.
Note that in a closed system, any irreversible process breaks time invariance in a subtle way, since energy may still be a conserved quantity although entropy is not (it increases). This however does not constitute an exception of Noether's theorem.
