Noether's theorem and "translations" of the Hamiltonian function In a nutshell, Noether's theorem states that for every continuous symmetry a corresponding conserved quantity exists.
Now, the Hamiltonian equations of motion (let's talk about a classical system here) are invarinat under addition of any constant to the Hamiltonian function
$$H\rightarrow H+ \text{const.}$$
Is there a corresponding conserved quantity?
 A: Nope, this operation is not a symmetry in the physics sense. A symmetry transformation is a transformation that changes or mixes the values of the basic degrees of freedom such as positions and momenta $x(t)$, $p(t)$ in mechanics or the values of fields such as $\vec E(\vec x,t)$ and $\vec B(\vec x,t)$ in electromagnetism.
The Hamiltonian is not an independent variable or a basic degree of freedom; it is a function of them. You're not changing the values of any quantities that evolve with time; instead, you're changing some formulae for derived and in principle unnecessary auxiliary objects in the theory (in this case the Hamiltonian), claiming that other formulae are preserved. This is not a symmetry so there is no conservation law associated with this operation. You're not "rotating" real objects which is what symmetry transformations should do: you're just redefining auxiliary, derived objects on the paper.
Incidentally, the operation you mention fails to be "harmless" in general relativity because the energy is a source of gravitational field - curvature of space - so if you move it by a constant, you do change physics.
A: What you're describing is not a symmetry in the sense of Noether's theorem of the (classical i.e. non relativistic and non quantum) system, but it is a symmetry of our description of the system.
A good example is provided when the Hamiltonian can be written as $H=T+V$ where $T$ is a kinetic energy and $V$ is a potential energy. Modifying the Hamiltonian by adding a constant to the potential energy does not change the behavior of the system in either the classical case or in the non relativistic quantum case.
This sort of symmetry is called a "gauge symmetry" and is discussed, in the quantum mechanics case, at length in Sakurai's quantum textbook "Modern Quantum Mechanics":
http://www.amazon.com/Modern-Quantum-Mechanics-2nd-Sakurai/dp/0805382917
See section (2.6) page 123, "Potentials and Gauge Transformations", which describes the difference between the classical case, where changing the potential has no physical significance, and the quantum case, where changing the potential changes the phase, but doesn't result in physically observable consequences.
A: Noether's theorem is that for every (differentiable) symmetry of the action (integral of the Lagrangian) there is a conserved quantity.  It does not directly say anything about symmetries of the equations of motion, nor about the Hamiltonian.
