Why is Noether's theorem important

Can quantum effects avoid Noether's theorem?

It is important to distinguish between local and global symmetries. "Noether's theorem" usually refers to her theorem that any continuous global Symmetry corresponds to a conserved stream. However, the proof of the theorem requires the use of the classical equations of motion, so that it does not hold in the quantum case. (More precisely, there are field configurations in which the charge is not retained, which contribute to the path integral.) As Aaron points out, quantum anomalies can break the classical symmetry and lead to the conserved current not being maintained (e.g. vacuum Maxwell's equations have a conformal symmetry, which is anomalous in the QED and therefore does not apply.

Local (or "gauge") symmetries, however, are a completely different story. These symmetries also apply without assuming the classical equations of motion (ie both "on-shell" and "off-shell"). All field configurations that contribute to the path integral take these symmetries into account. Gauge symmetries (like the U (1) example you mention) also correspond to conserved quantities, but these remain always conserved, even taking into account quantum fluctuations. (Gauge symmetries can also be anomalous, but rather than just leading to a violation of the conservation of the conserved set, anomalous gauge symmetries prevent you from consistently quantizing your theory in the first place, so "breaking" the whole theory.)

ACuriousMind ♦

Their language is dangerously imprecise: all symmetries hold outside the shell, that all variations of action on the shell vanish, is the definition of "on the shell" and therefore "on-shell symmetry" is not an interesting concept. "All field configurations that contribute to the path integral take these symmetries into account." is also a strange statement - what does it mean for a Field configuration to "respect a symmetry"?