The game makes it blatantly obvious that Fang and Vanille have a close bond. Just about everything they do is for the other's sake, whether it's Vanille's lying about her memories in hopes that the past will not repeat itself or Fang's decision to become Ragnarok to save Vanille from the same fate. These are hardly small things; rather, they're serious decisions made in order to protect each other. The question is, then, just how close are they? Are they simply close friends, or does it go deeper than that?
The evidence — of which there is a great deal in the form of their interactions — does not point toward one answer or the other, leaving it up to the player to decide. I'm of the persuasion that there is a close friendship between them but also a romantic relationship. It's really not a stretch to conclude that based on what goes on between them.
I'll tear down the sky if it'll save her.
With these words, Fang expresses quite clearly to Lightning (and the player) just how much Vanille means to her. Ironically, she's also making a reference to their past unknowingly: Fang was the one to become Ragnarok and scarred Cocoon in an attempt to "tear down the sky."
Allusions aside, these are heavy words, and Fang means every one. Vanille obvious matters enough to her that she'll go to any lengths to protect her. You could say it's simply indicative of Fang's intense personality, and it certainly is, but it's not exactly a stretch to see more into it.
Another thing worth mentioning is how Fang refers to Vanille: as her "partner." Not her friend, not as a sister, but as a partner. That, too, can be taken as simply platonic, but it also has the connotation of being a word to refer to a significant other. Again, there's no way to say for sure, but it's certainly a possibility.
One of the things that sets FFXIII apart is its extremely high-quality graphics and the subsequent physical motions shown. It's a lot easier to observe the characters' physical mannerisms than in previous games, far more along the lines of a movie than a novel in expressing what characters are doing.
The way Fang and Vanille move around each other shows the level of comfort they have around one another. When they are reunited, they immediately hug. From that point on, Fang doesn't hesitate to lift up Vanille's skirt to check her brand at any time, heedless of anyone else. Vanille seems embarrassed at Fang's forwardness, but not overly so, perhaps implying that she's comfortable with Fang doing it. Fang also hugs Vanille after she comes clean about lying to her and fighting her Eidolon, offering her comfort when she needs it most.
Also, and because it's obligatory, the boobgrab.
Take from that what you will.
The finale of the game focuses heavily on Fang and Vanille, which is fitting as most of the other characters' conflicts have been resolved. Fang becomes Ragnarok for Vanille's sake, though it's Vanille that shows her it's not necessary. And in the end, the two of them realize what they have to do to complete Vanille's Focus: save Cocoon from falling to Pulse by becoming Ragnarok together and crystallizing themselves to protect it forevermore.
There's no hesitation in what they do in order to save Cocoon, nor in resigning themselves to return to the way they were in the beginning: in eternal crystal slumber together. If you're looking at it as a shipper, it's downright romantic (albeit tragic).
In conclusion, no matter how you look at it, there's no denying Fang and Vanille have a close bond. Whatever there is between them, it's obvious that they care deeply about each other. I like to think they're in a mutual relationship, but it's also entirely possible that they have unspoken attraction to each other. Really, whatever you see in their relationship is entirely up to you. And in truth, that's why I like it so much: nothing is lost from it if you don't see it as romantic. They share a beautiful relationship either way. ♥