In field theories like GR, energy is generator of time translations. It's conserved if time translations leave the system invariant, but this only occurs in very special cases in general relativity (static spacetimes). So in general, in Gr, energy is not conserved. Same thing goes for momentum, which generates spatial translations.
Another question on a tangentially related topic: Could you perhaps shed more light on why there so much pessimistic skepticism towards any claims of working reaction-less propulsion drives (violating conservation of momentum), for example?
I understand the experience-based reasons for skepticism (since there have been many claims that don’t replicate), but why do there also seem to be so many physicists claiming with 100% confidence that no such thing could ever work, according to current physics theory?
I’m referring for example to skepticism about things like “EMDrive” from a theoretical perspective (e.g. the posts from physicists here: https://www.reddit.com/r/EmDrive/) as opposed to focusing on experimental failures to replicate.
Because it would violate conservation of momentum. Now wait a minute, you might think, haven't we just learned that such conservation laws don't hold in GR?
It's more complicated than that.
An analogy would be doing mechanics in a rotating reference frame. To make this work out, you have to account for things like centrifugal forces, with energy contributions that come 'out of nowhere' - as in, if you go to the non-rotating, inertial frame, they vanish: There's no real physical source providing that energy.
Something similar happens in general relativity, where you can get conservation laws via Noether's second theorem even if your spacetime is not static (or rather stationary, to be pedantic). You will generally get a contribution from the gravitational field, which cannot be localized, but also - in contrast to the previous example - cannot be made to vanish because global inertial frames do not exist in general.
One should perhaps also mention that reactionless motion supposedly is possible in classical physics: It's been claimed that a body can change its position without ejecting any propulsive mass through a series of deformations in an inhomogeneous gravitational field. There's a cute name for this, 'swimming in spacetime'.
The fact that energy isn't conserved for a fixed metric is basically just what makes flybys capable of increasing the kinetic energy of a projectile.