From Wikipedia,
"The local energy conservation in quantum field theory is ensured by the quantum Noether's theorem for the energy-momentum tensor operator. Thus energy is conserved by the normal unitary evolution of a quantum system."
I know that it is the case in GR that conservation of energy and other conservation laws are relegated to being local only I thought this wasn't the case in quantum field theory.
Yes the stress energy tensor operator, also in curved spacetime, satisfies the local conservation equation in a distributional sense $$\nabla_a :\hat{T}^{ab}:_{\omega}(x)=0\:,\tag{1}$$ where the "normal order" is referred to a (vacuum) Gaussian (Hadamard) state $\omega$.
I wrote a work on that many years ago which proves that also at the level of *-algebras of operators (I considered only the KG field, but the procedures can be extended to all types of fields).
Theorem 2.1 (with $V' \equiv 0$) in https://link.springer.com/content/pdf/10.1007/s00220-002-0702-7.pdf
There is also an arXives version.
(1) implies a global conservation law in the presence of a timelike Killing field $K$ $$\int_\Sigma \langle \Psi| :\hat{T}^{ab}:(x) \Psi \rangle K_a n_b d\Sigma(x) = constant$$ where $n$ is the unit vector normal to the Cauchy surface $\Sigma$, for every state $\Psi$ on a suitable domain in the Hilbert space of the GNS representation of $\omega$. It is not obvious that the integral is finite.
Concerning the issue of conservation laws in GR, they exist when Killing vector fields exist. Properly speaking, (1) is not a conservation law as it stands: in the sense of the equivalence principle is, but not in the sense of the Noether theorem.
However it becomes a conservaqtion law in proper sense if there is a Killing vector field. Let us prove it.
A conservation law is of the form $$\nabla_a J^a =0\tag{2}. $$ If this identity holds, the divergence theorem proves that $$\int_\Sigma J^a n_a d\Sigma = constant \tag{2'}$$ on Cauchy surfaces $\Sigma$, where $n$ is the unit vector normal to $\Sigma$. Obviously it is also necessary that the integral converges (Actually, even if it diverges, we may also consider it a global conservation law: the charge is aways infinite. It depends on personal taste.)
Equation $\nabla_a T^{ab}=0$ does not automatically imply (2). However, if there is a Killing field $K$, so that the Killing equation is valid
$$\nabla_a K_b + \nabla_b K_a =0 \tag{3}\:,$$
then $J^a:= K_bT^{ba}$ (where $T^{ab}=T^{ba}$) satisfies (2).
$$\nabla_a (K_bT^{ba}) = (\nabla_a K_b) T^{ba} + K_b \nabla_aT^{ba} =
\frac{1}{2}(\nabla_a K_b + \nabla_b K_a ) \: T^{ba} +0 =0\:.$$
