See here for information on the chronology protection conjecture.
If somehow, it is only possible to create one alcubierre drive, and it can never turn around, then you won't get closed timelike curves. Otherwise, any construction is going to have them.
The reason is that you can "zoom out" far enough that the distortions to spacetime caused by the drive are no longer present, and the person flying the drive then just looks like a spacelike curve in the spacetime. At that point, you're just travelling along spacelike curves, and all the special-relativistic causality objections apply.
We can't say much about closed timelike curves with any certainty; they are an artefact of the existence of solutions to the general relativity equations which allow them. It is possible (and quite a few physicists believe this) that a theory of quantum gravity may preclude CTC from occurring, or that CTC may occur but the information might be censored by an event horizon. It is also possible that CTC may occur but only Universes which fail to contain paradoxes exist, in which case it is unlikely that humans will ever have the ability to create CTC.
Using only Einstein's general relativity, anything that allows information to travel faster than $c$ could create a closed timelike curve and potentially the paradoxes which accompany this sort of thing. So, no, you can't have an Alcubierre drive without the ability to create closed timelike curves, even if you only travel in one direction (no offense to @JerrySchirmer but we part ways on this fact). Consider two events A and B, where in all subluminal frames of reference A barely causes B by communication at speed $c$. You're heading FTL in the opposite direction: in your frame of reference B actually precedes A. Now you can contact A after B happens as you fly by (to prevent B from ever happening). Whoops.
I've looked into this, and Jerry is indeed correct, at least in part: one-way FTL travel looks paradoxical to the traveler, but is otherwise paradox-free. However, I believe the drive would need to create gravity wells on the order of a black hole to function properly which means any object in its vicinity (e.g. a mass that gets clipped by the bubble as it passes) could create a closed timelike curve. So no, it can't directly cause a paradox; but you can't prevent the possibility of indirect paradoxes caused by the wake of the device.
This took me a while to figure out in part because what an Alcubierre drive would actually do is inconsistent with general relativity, and GR principles are what the device is based upon. This leaves you with an ill-defined object. The main problem is that there exists no fixed "background" to spacetime, so the only way the bubble can move something is by causing it to "fall" in one direction (a relativistic hill behind you, a gap before you, and the bubble moves with you as you fall). This means it grabs hold of you by accelerating you in one direction; and if the bubble ever goes FTL, it must lose its cargo in the process (you'll slide backwards over the hill). This implies that the only objects that can go FTL inside the bubble are objects which were already going FTL to begin with, in which case you didn't need the bubble in the first place.
(Not to mention, it breaks enough conservation laws to make Emmy Noether spin in her grave.)