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.
Best Answer
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)