Why should one expect closed timelike curves to be impossible in quantum gravity? From the Wikipedia article, it seems that physicists tend to view closed timelike curves as an undesirable attribute of a solution to the Einstein Field Equations. Hawking formulated the Chronology protection conjecture, which I understand essentially to mean that we expect a theory of quantum gravity to rule out closed timelike curves.
I am well-aware that the existence of closed timelike curves implies that time travel is technically possible, but this argument for why they should not exist isn't convincing to me. For one, if the minimal length of any closed timelike curve is rather large, time travel would be at least infeasable. Furthermore, this is essentially a philosophical argument, which is based, at least in part, on our desire to retain causality in studying the large scale structure of the universe.
So far, the best argument I've heard against CTCs is that the 2nd law of thermodynamics wouldn't seem to have a meaningful interpretation in such a universe, but this isn't totally convincing. A good answer to this question would be some form of mathematical heuristic showing that in certain naive ways of combining quantum mechanics and gravity, CTCs are at least implausible in some way. Essentially, I'm trying to find any kind of an argument in favor of Hawking's conjecture which is not mostly philosophical.  I realize that such an argument may not exist (especially since no real theory of quantum gravity exists), so other consequences of the (non)existence of CTCs would be helpful.
 A: It is widely believed (but it has not been rigurously proved so far) that to sustain stable closed timelike curves, you need copious (i.e of the order of the total mass of the universe) quantities of exotic matter. Exotic matter is just a generic term to describe matter for which the stress-energy tensor satisfies
$$ g^{\mu \nu} T_{\mu \nu} < 0 $$
That is, exotic matter violates the Strong Energy Condition, which is known to hold in all known quantum physical theories.
This applies not only to CTC, but as well as stable wormholes, warp drives, or anything actually fun. This is in itself, outstanding evidence of the catholic nature of God, since interestellar travel would raise the need for some akward explaining from the Pope (i know, BS, but let me have my punchline please)
A: Closed timelike curves can be used to create paradoxes! That's the reason why they are a problem - because you might go back and kill your grandfather. This is not just a "philosophical" argument, it's a matter of logic. 
The principle that reality is not self-contradictory - i.e., that there are no real paradoxes, only apparent paradoxes - allows you to deduce that, in reality, CTCs can't be used to create paradoxes. So either there are no CTCs, or they are "harmless" in some way. 
Logic can get you that far: you should expect CTCs to be impossible or to be harmless. But your question goes further and asks why we should specifically expect them to be impossible. Well, maybe we shouldn't. CTCs can show up in anti de Sitter space; they might show up in a gravitational path integral; there might be a CTC at the big bang (as suggested by Gott and Li); there might be a giant CTC connecting cosmological future and past (as suggested by Gödel). I'm not aware of a knockdown technical argument (i.e. a totally robust physics argument) against any of these. 
A: It's certainly possible that just as the holographic principle can save unitarity for black holes, that some generalization of the holographic principle coupled with cosmic censorship for closed timelike curves can save unitarity.
At any rate, the interior of a time machine is only real to the extent that memories and records of the interior can get out.
A: These are some notes to complement previous answers.
Your concerns are sound: a rigorous definite way to rule out CTCs has not been found. What we have is arguments (and quite nice looking ones) to illustrate that every known universe with CTCs looks unphysical.
Second, there are two nicely-written pedagogical letters written by Kip Thorne addressing your question [1],[2]. They mainly focus on physical aspects of the known CTC solutions, and three popular mechanisms that could prevent CTCs: violation of the averaged null energy conditions (the first argument cited in the post), classical instabilities of chronology horizons, and quantum field instabilities (following the notation of [2], section 4). Although he does not seem to believe in CTCs personally, at the end of [2] he states that this is still an open question:

It may turn out that on macroscopic lenghscales chronology is not always protected, and even if chronology is protected macroscopically, quantum gravity may well give finite amplitudes for microscopic spacetime histories with CTCs [29].
[29] Friedman J 1992 in Proceedings of the 4th Canadian Conf. on General Relativity and Relativistic Astrophysics eds G Kunstatter et al (Singapore: Word Scientific) pp. 183-199.

Finally, regarding the argument against CTCs that uses logical paradoxes, which has already appeared in the post: it is not clear to many people whether CTCs inevitably lead to causal paradoxes. Several studies have pointed out that causal-paradoxes of time travel could disappear once one takes quantum mechanical effects; or maybe their meaning could simply change [3],[4],[5],[6]. For instance, in the framework used in the first reference the grandfather's paradox does not violate causality. In connection with this, although it is known that some of these models of CTCs [7],[8] lead to counter-intuitive collapes of computational complexity classes, this is not exactly the same as a causal paradox.
