Do any quantum gravity theories deal with closed timelike curves?
As far as I'm aware, there are no quantum gravity theories that deal directly with closed timelike curves.
As far as I know that's correct.
Some of them (like canonical QG, CDT and LQG) forbid them outright, others merely seem to not discuss the topic.
For good reason: closed timelike curves are not what most people think. There's a number of issues within general relativity which are magnified in quantum-gravity theories, some of which have been kicking around for decades, and none of which have proven at all successful. For good reason, but that's one for another day. What isn't, is that a closed timelike curve is an abstract thing. The important thing to understand is that you don't travel around it. Instead blood moves through your body, electrochemical signals move through your brain, the Earth moves, light moves through space, and so on. Your worldline is an abstract imaginary depiction of motion through space in a static 3+1 dimensional "block universe" called spacetime. It's like I throw a red ball, and you film it then you develop the film and cut it up into individual frames and form them into a block. The red streak in the block is akin to the ball's world-line. But the ball is not moving up that red streak, and in similar vein you don't travel along your worldline. Objects don't move through spacetime, objects move through space. The misconception that they do move through spacetime along a worldline probably stems from Wheeler. See A World Without Time: The Forgotten Legacy of Gödel and Einstein, where author Palle Yourgrau says Wheeler conflated a circle with a cycle, "precisely missing the force of Gödel's conclusion":
I've found quite a variety of QFT behaviour in classical spacetimes with closed timelike curves, including string theory in a CTC background, but I can't really think of any paper where the metric wavefunction (or sum of metric histories or whatever else) might run over acausal states.
Nor can I.
The obvious candidate for this would be one of the variant of path integrals like Euclidian gravity, Lorentzian gravity, Regge calculus, etc. But there seem to always be this assumption that the boundary conditions (if present) will always be on spacelike hypersurfaces, which, while it does not make it impossible to have closed timelike curves on such spaces, certainly restricts their numbers (I suspect that CTCs in such a case only arises from the topology and not the metric).
We live in a world of space and motion. Worldlines are abstract things that don't actually exist, nor do light cones, nor do spacelike hypersurfaces, and nor do closed time-like curves. Try to think what a CTC really is: it isn't something you can travel around, GR doesn't allow for time travel. If your world-line really was some closed timelike curve that was 24 hours long, your life lasts for 24 hours, and it is causeless. It's like some SciFi Mayfly Day. You are born from an egg, you live for a day, you lay the egg, you die. And that's it. There is no repetition, it isn't like Groundhog Day, because you don't travel along your worldline.
String theory might also work out, as I am not aware of any theorems forbidding the strings to reduce to CTC solutions in the classical limit, but I do not know that much about string theory unfortunately.
Forget string theory. It's a mathematical fantasy that has gone nowhere for fifty years.
Are there any papers discussing such topics? Are they even possible in the context of any of those theories as we currently understand them?
You can always find something, see the arXiv for example. But I wouldn't waste my time with them if I were you. Instead you'd be better off spending more time on GR and understanding why CTCs are science fiction. Pay more attention to the Einstein digital papers, and less attention to Kip Thorne.