Does this violate local energy conservation?

Teleporting Energy

And if so, how would this affect things like gravitational physics?

It *might*.  It seems that with quantum entanglement, you have to give up *something* you normally want, like the principle of local causality, but you seem to have your choice as to which of two or three principles to give up.  David Bohm's pilot wave version of quantum mechanics gives up one; most people seem more inclined to give up the other; but I do not recall what the choices are.  Sorry.

@RainbowRising, well done for drawing attention to the inaccurate leading paragraph. However, I see no reason to disbelieve the original source.

Firstly, the original paper explicitly states that local energy conservation is not broken, making my title rather foolish!

The novel transportation of energy from one place to another can only occur when information is transferred between the two locations using other means, at a speed not exceeding that of light. Eg we send a radio signal to Alpha Centauri, and when it arrives, a parcel of energy (greater than that in the radio signal) mysteriously appears there as well. The almost magical thing is that the energy travels from one place to another without passing through any intervening points, which is radical enough to justify the last clause you quoted.

However, one of the other parts of the original paper that I can understand is that the energy teleported is limited to that which can be "hidden" in the zero point fluctuations of the second quantum system. So if I understand correctly, what you have is a system with some uncertainty in its energy being resolved into one which has a fixed amount of energy (perhaps the top of the possible range of energies). I presume it is the ability to pick which end of the range the energy gets resolved to that justifies calling it teleportation of energy.