In my field theory class we recently derived Noether's theorem: We consider a infinitessimal transformation $\phi \to \phi + \epsilon \,\delta\phi$ of our field which preserves action i. e. $\delta S = 0$. This last condition is supposedly equivalent to $\delta \mathcal{L}$ being a divergence, i. e. $\delta \mathcal{L} = \epsilon\,\partial_\mu I^\mu$. Then you can expand \begin{align} \delta \mathcal{L} &= \frac{\partial \mathcal{L}}{\partial \phi} \epsilon \, \delta \phi + \frac{\partial\mathcal{L}}{\partial (\partial_\mu \phi)} \epsilon \partial_\mu(\delta\phi) \\ &= \left( - \partial_\mu \frac{\partial\mathcal{L}}{\partial (\partial_\mu \phi)} + \frac{\partial \mathcal{L}}{\partial \phi}\right) \epsilon \, \delta \phi + \partial_\mu \left( \frac{\partial\mathcal{L}}{\partial (\partial_\mu \phi)} \delta\phi\right) \epsilon \end{align}
The first term vanishes by the eqns. of motion and so we get $$ \partial_\mu \left( \frac{\partial\mathcal{L}}{\partial (\partial_\mu \phi)} \delta\phi - I^\mu\right) = 0 $$ i.e. $\frac{\partial\mathcal{L}}{\partial (\partial_\mu \phi)} \delta\phi - I^\mu$ is a conserved current.
Two things confuse me here:
We didn't use that the transformation is supposed to be a symmetry of the system (that is $\delta S = 0$). In a point-particle setting (that is a field that only depends on time) we can have $L = T - V$ with $\partial V / \partial q \neq 0$, but we can look at a translation $q \to q + \epsilon$ which gives us $$ L \to L + \epsilon \frac{\partial}{\partial t} \left( - t \frac{\partial V}{\partial q} \right) $$ so $L$ is only changed up to a "divergence" and the resulting conserved quantity is $p + t \frac{\partial V}{\partial q}$, which is indeed conserved, even though our system was not space-homogeneous.
The other source of confusion (which I guess is related to the first) is this argument that $\delta S = 0$ is equivalent to $\delta \mathcal{L} = \epsilon \partial_\mu I^\mu$. In $\mathbb{R}^n$, every scalar function can be written as a divergence, so this doesn't seem to add up. Is $I^\mu$ maybe supposed to be a function of the fields only?
