White Collar/WMG

Neal used to have no problem with using guns until some sort of traumatic experience gave him an aversion.
It's obvious. You don't get to be that good a shot unless you've had a lot of practice, and you have to be able to use a gun to get that practice.
 * Not necessarily a traumatic experience - he could just decide not to use guns as a matter of preference. First off, he doesn't exhibit any psychological signs of aversion to picking up and using a gun (quivering hands, shortness of breath, etc), and is in fact quite eager with his suggestion that he be combat-trained by the FBI. Secondly, it seems his dislike of guns is more a matter of style ("murder is not an intellectual pursuit"), as he would rather use his mind, and seems to consider violence beneath him (except for when it comes to finding out about Kate).
 * Probably has to do with his daddy issues. As a kid he wanted to be a cop like his father, then when he found out his father was crooked he distanced himself from everything to do with law enforcement in disgust, including guns. This is also why he became a (nonviolent) criminal.
 * It probably also comes from the fact that he's white collar and prefers to keep it that way. Law enforcement are generally much harsher with people that use violence. By being so openly non-gun using, Neal can limit the risk of law enforcement attempting to actually shoot him. And on a Chessmaster level, by limiting the level of force that will be placed on him, Neal can also ensure lighter security on him when in custody and limit options when being pursued.
 * It seems from his attitude just as likely that he simply doesn't like hurting or killing people.

Kate is secretly with the FBI.
She's either from another branch, or in so deep, even the FBI doesn't know she's one of them.

Kate is secretly with the NSA.
If there is any government agency that would infiltrate the FBI, then the NSA is it.
 * It doesn't have to be. The intelligence community is made up of more than just the FBI, CIA, and NSA. There are in fact something like 14 different agencies within it. Counting each branch of the military, that's only 7. It would be more mysterious to use one of the agencies that people -don't- know about.
 * 16 in fact. They are (some with heavily abbreviated or shortened names): CIA (independent), Air Force Intelligence, Army Intelligence, Defense Intelligence, Marine Intelligence, National Geo-Spatial Intelligence, National Reconnaissance Office, NSA, Office of Naval Intelligence (all DOD), Office of Intelligence & Counter Intelligence (Department of Energy), Office of Intelligence and Analysis, Coast Guard Intelligence (DHS), FBI, DEA (Justice Dept), Bureau of Intelligence and Research (State Dept), Office of Terrorism and Financial Intelligence (Treasury). To be fair to those not called the FBI, CIA, NSA, most of the others don't sound nearly as sinister. And those not listed like the Special Activities Division (part of the National Clandestine Service) are just part of other organizations anyway (CIA for the example).

Kate is the real mastermind
She's been playing on Neal's emotions to get the music box, and then she faked her death, knowing it was the only way he would stop looking for her.

Kate is related to Damien Moreau, the guy the Leverage team is supposed to take down.
We know Kate's father's name, so I'm guessing brother and sister. Mostly because I would love to see Neal team up with Nate Ford's crew.

June's husband was high up member of the Mafia
Let's see, obsecenly wealthy in New York, impecable style and love of suits and fedoras, ex-convict, good friends with the ? It adds up.

Kate is not dead.
The woman on the plane was a decoy. Either Kate is secretly evil, or there's a Big Bad who, for reasons as-yet unknown (probably having to do with the music box), wants Neal to think that Kate is dead.
 * Seems significantly more likely given the season finale.
 * Sorry, but according to Word of God, they waffled about this for awhile before deciding to yield to the fans and making her Killed Off for Real.

The man Burke and Diana chased down at the beginning of "Need to Know" was Neal.
He's probably meeting with Fowler to get information on who killed Kate. It also explains why he was so careful never to let them see his face.
 * Debunked: It was Julian Larson, a guy Fowler trained with back during his spec-op days.

Mozzie is actually Martin Lloyd
It explains why he's using an alias and is extraordinarily paranoid about the government.

Julian Larson works for John Barret.
Giant conspiracy? Ex-military? This is starting to look like another big conspiracy. Plus, we've had more and more hints that a crossover might be in the future.

Neal is Bryce Larkin's new life (second? third?) from Chuck
Well, at least the actor's. And maybe the bringing back to life causes a loss of kung-fu ability, but it would explain Neal's skill set and dislike of guns, and hangup about certain women.

Peter is the same character Tim DeKay played in Swordfish
He worked for the FBI and was never named so it's not inconceivable.

Vincent Adler is a Nazi
Or rather, his family were Nazis, and he wants to claim whatever the music box points to, which is a hidden cache of Nazi Gold or evidence of his family's crimes. The china found in "Burke's Seven" introduces the Nazi connection, Adler is a common German surname, and there are mentions of Argentina (where all Those Wacky Nazis went to hide) in both "Burke's Seven" and "Forging Bonds".

Neal Caffrey is an alias
Do you really think that Peter would not have known that Neal didn't graduate high school if he knew Neal's real name?
 * Season 3 seems to indicate
 * Not necessarily. The comment that  could theoretically imply that he came up with the name himself, but this troper finds it more likely that Neal feels   because it is his real name. YMMV, of course.

The world of White collar is purgatory, and Neal and Peter are incarnations of Kafka and Poe respectively
This is an afterlife in which people live out their deepest dreams. Neal and Peter represent all of the things that Kafka and Poe wish they were, but aren't. And yet, some of the aspects of their original personalities come through, like Peter's mustache. Neal (or Kafka), in particular has difficulty fully believing in his own fantasy, which is why he can't form a lasting relationship, has trouble finishing anything, and is mostly estranged from his family. Only when they both work through the remainder of they're demons will they finally be able to move on.
 * Neal easily gets along with people, especially women, has tons of natural confidence, and is successful at what he does. In his fantasy, Mozzie represents his country doctor uncle, one of the few people he got along with, and Peter represents Max Brod, the only person who ever believed in him.
 * Peter has a long, stable relationship with his wife who has not died of a horrible disease, a position of power and influence, and is certain in his worldview.

Kate and Elizabeth are related somehow
Just look at them! It would be a case of Separated At Birth Casting, because they look like they could be related.

Peter/Neal will eventually become canon.
Either that, or Neal/Peter/Elizabeth will become canon. Jeff Eastin is aware of the Ho Yay (and he seems to be encouraging it), so I don't see why not.

Neal will stay with the FBI after his deal is done.
In Payback, it's outright stated that Neal really wants that normal life. You can see a quick moment of envy when he watches Peter reunite with Elizabeth. And both he and the FBI quickly work together during the episode; even Keller notes that Neal is sounding more like the FBI and less like a con at the end.
 * Seems less so after the season 3 intro since Neal doesn't skip a beat in making an escape plan with his plunder.
 * Near the end of "Deadline", there's a glimmer of hope that Neal will stay when Peter says that Neal is officially part of the family. After he says that, there's a look on Neal's face that shows that he might not want to leave. And if you pay attention during "On Guard", Neal does seem a bit hesitant on leaving.
 * At the end of "Taking Account" Neal outright tells Mozzie that he would like to stay in New York and Mozzie has to remind him that it would never work out as long as they have the treasure.
 * And now that his commutation hearing is upcoming, Neal actively discussed the idea of becomming a paid consultant for the FBI. At the very least, this option is forefront on people's minds.

Neal will give up the U-Boat treasure in order to stay in New York with his new friends and new girlfriend.
It's constantly lampshaded that Neal cannot stay in New York if he wants to sell the treasure and live off the proceeds. Neal will decide that the treasure is not worth it and voluntarily give it to the authorities.
 * Either that, or he might leave with the U-Boat treasure, realize he was better off in New York, and come back. Might make for a good season finale and Tear Jerker.

Keller will be taken down for good in the remaining episodes of season 3.
, and most likely Peter will go all out in. Either Keller will finally be arrested or Peter will (most likely) try to kill Keller once he finds him and Neal will have to try to talk him out of it as a Call Back to "Point Blank".
 * Finally be arrested? He's been arrested, what, twice now? He just keeps escaping.
 * I meant that he would be locked away for good, put in a higher security prison or something.
 * He's the Joker to Peter's Batman, effectively.
 * Who does that make Neal? Catwoman? (Oh god oh god I just know I gave people ideas ... )

Peter is Neal from the future

 * Peter and Neal are My Future Self and Me. Peter was Neal Caffrey, and since he never was caught, he got too confident and pulled off bigger and bigger heists, until pulling off the U-Boat heist, beating Adler and Keller to it. Doing so angered them, and in revenge killed all the people he cared about (Moz, Kate, Alex). Filled with regret and with no one to turn to, he happens to have stolen a device with Applied Phlebotinum that sent him back 15 years creating an Alternate Timeline. Being able to forge a new identity, he decided to start over as Peter Burke and dedicate himself to stopping his younger self from making the same mistakes. In order to do this, he forges a resume and gets into the FBI, working his way into White Collar division in order to capture Neal. In the meantime, he sees someone who reminds him of Kate and decides to put an FBI tail on her, only to discover it's someone named Elizabeth. Peter is the only one to catch Neal because Peter knows exactly what Neal would do. Of course, he's not always right, because now with Peter in the timeline, Neal will react differently. When Peter catches Neal the second time, Peter knows Neal will be good with White Collar crime division because, well, they are the same person. Sadly, he was still not able to save Kate because in the original timeline, Adler killed Kate several years later.

Elizabeth is going to be a Badass Damsel.

 * The promos for the new episodes show her trying to break out of a room with a chair.
 * If she doesn't punch Keller's lights out at some point, I am going to be very disappointed.
 * Confirmed!

Neal will kill somebody in the new episode.
He is shown in a certain promo bloodied with a gun aimed at someone off-screen. It's most likely Keller he's aiming at.
 * Half Jossed; he shot in the leg.

Elizabeth is Nate Ford's half sister.

 * Tom Skerrit plays both of their fathers, and there's the mention on John Rogers' blog of "meeting Nate's sister".