Energy is the conserved current associated with the time-symmetry in time-independent Lagrangians. For problems whose Lagrangians lack that symmetry, there will be no such conserved current. Attempting to define one will produce a time-varying quantity instead.
In Galilean physics time-asymmetric Lagrangians are associated with problems that include e.g. friction terms. In these cases, though, it is always possible to consider "larger" problems that account for the friction dynamically rather than statistically, so energy conservation can always be recovered in principle. This is because the background manifold upon which the problem lives is itself time-symmetric.
In general relativity, however, there is no guarantee that the background will be time-symmetric. In other words, there are perfectly good solutions that lack timelike Killing vectors, because the gravitational field varies in time. In such cases even defining energy is ambiguous: there is no symmetry to compare to. One could in some cases define the energy in terms of a timelike Killing vector associated with a related time-symmetric spacetime, but in this case the energy won't be conserved.
The basic implication of this is that energy is not conserved in any spacetime with a time-varying gravitational field. Strictly speaking, that means energy conservation fails anytime anything moves such that its quadrupole moment varies (of course, in almost all practically interesting cases the magnitude of the failure is small).
We normally deal with asymptotically flat problems, however. In that case, it is possible to recover some notion of energy by using the timelike Killing vector at spatial infinity to define a total energy for the whole spacetime. This allows one to say things like "gravitational radiation has carried $X$ solar masses away from a black hole merger", by approximately treating the binary black hole system as an entire spacetime. However, one cannot pinpoint the location of the GW energy flux to specific points of the binary in such a way that the energy accounting works out, because the timelike Killing vector does not exist at those points. If one tried to do that they would find the specific points had "energy" which was not accounted for by the GW flux.
In other words, the energy of a gravitationally-radiating object is in some sense conserved globally but not locally. This is basically because the energy density in the gravitational field itself cannot be defined in a covariant way. But local conservation of energy is what is really important: if I were to show you a perpetual motion machine here on Earth, it would be small comfort to learn that energy somewhere in the Andromeda Galaxy were decreasing to power it.
Therefore, here is my first example of a failure of energy conservation:
-Anything radiates gravitational waves for any reason.
The most important example of a problem which is not asymptotically stationary is the FLRW expanding Universe. There is no timelike Killing vector anywhere and thus no conserved energy, even globally. If we consider a volume of space at two times $t_0$ and $t_f$, $t_f > t_0$, filled with, say, light, the total energy within the volume at $t_f$ will be less than that at $t_0$, since the photons have been redshifted. If there's a cosmological constant, the energy will actually increase: there is now "more space" to fill with cosmological constant.
Note this is importantly different from a photon ascending through a static gravitational well. While the photon is redshifted in this case, one can reconcile the lost energy in terms of the static gravitational potential of the Schwarzschild solution. But no such constant potential exists in the FLRW universe.
Edit: OP views the FLRW Universe as an invalid solution to the EFEs, since it is homogeneous and isotropic. The arguments given above apply, however, equally well to the Kasner metric
$ds^2 = -dt^2 + \sum_{j=1}^{D-1} t^{2p_j} [dx^j]^2$
which describes an expanding/contracting family of universes which are neither homogeneous nor isotropic (the expansion rate is different in different directions).
Edit2: OPs suspicion of the FLRW solution appears to be motivated by the assumption it describes the "entire Universe". It is important to point out that while this is certainly the most important application of the FLRW solution, it is not the only one. In particular, the FLRW solution can be smoothly joined to the Schwarzschild solution in order to describe a finite ball of homogenous and isotropic pressureless matter, which will expand or contract. Of course real matter has pressure, but up to this approximation this is in principle something you could build in your living room. Within the ball, energy will not be conserved.
Edit 3: Symbolic demonstration of the dependence of energy conservation on symmetry (following http://www.blau.itp.unibe.ch/newlecturesGR.pdf).
The stress-energy conservation law
$\nabla_\mu T^{\mu \nu} = 0$
expands to
$g^{-1/2} \partial_\mu (g^{1/2} T^{\mu \nu} ) + \Gamma^\nu_{\mu \lambda} T^{\mu \lambda} = 0.$
The energy-momentum density $T^{\mu 0}$ is neither conserved nor covariant, due to the Christoffel term above (the Christoffel symbols are not a tensor). We can attempt to form a conserved current by contracting with some vector $V^\lambda$; thus we seek $V^\lambda$ such that
$\nabla_\mu(T^{\mu}_{\lambda} V^\lambda) = 0$.
This will correspond to energy-momentum conservation if $V^\lambda$ is timelike. The above expands to
$\frac{1}{2} T^{\mu \nu} (\nabla_\mu V_\nu + \nabla_\nu V_\mu) = 0$.
So we will have energy conservation iff the bracketed term vanishes. The bracketed term is Killing's equation: it vanishes iff $V_\nu$ generates an isometry, i.e. if the Lie derivative of the metric along $V$ vanishes: $L_V g_{\mu\nu} = 0$. But no such $V_\nu$ exists in general.
To show energy is not conserved in a particular case, pick any non-stationary spacetime you like, and form the above current using any timelike vector field you like. You will not obtain a conserved quantity.
