Assume a gravitational field, with area A having some gravitational forces while area B having no gravitational force. The Lagrangian of particle moving along this field is obviously time invariant, but the mechanical energy (Hamiltonian) is by no means conserved if you go from region A to region B horizontally, moving vertically in region B, then go horizontally to region A. You gain energy after moving to the point with higher potential energy. This breaks conservation of energy but Noether's theorem fails to tell there either. I know as someone's answer under this question might say, such field would never be real because it's highly discontinuous (and thus not differentiable, so Lagrangian makes no sense), however you can always make the gravity continuously go to 0 and moves higher through space with 0 gravity then move back to points with non-zero gravity.
Same issue occurs with presence of friction, because the Lagrangian doesn't depend on time explicitly (there is no point to assume friction depends on time), yet with presence of friction, no conservation of mechanical energy would occur, even though Lagrangian stays the same over time.
I think both examples have issues with having non-conservative forces, because least action principle only applies with conservative forces, but if Noether's theorem only applies to systems with conservative forces, what's the point then as energy in that system is already conserved by the definition of potential and kinetic energy?