Obviously this happened with LotR to some extent - things were set up in the EEs for the next film - but with all the rewriting it does seems that the new material trumps continuity with TH...
Captain Salt has put together a list of the most notable discrepancies/confusing plot points...
1. The backstory is a bit of mess; we're two movies in and we don't know how Gandalf got the map and key from Thrain. The first film made it seem like the dwarves had assembled due to "signs" such as birds flying around, then sought out Gandalf's aide ("you asked me to find the 14th member of this company", Saruman's remark "if they had come to me",) while this film, while truer to the source material, shows otherwise. Also, as hutch pointed out, the Thorin-hunting story lacks consistency...were Azog/the Necromancer hunting Thorin so prevent him from "usurping" the dragon, and if so, why were one or the other doing so when Thorin was an aimless vagabond? If the Necromancer wants Thorin dead (it is he, not Azog who has plans to Erebor), why does he give up on the dwarves only to have Azog assign Bolg to to continue the hunt of his own accord? For that matter, how do Azog and his orcs transport to Dol Guldur seemingly instantaneously, while Bolg's company manages to make their way to the Woodland Realm by the following night? Also, there's no hint that Azog serves the Necromancer in Film 1 (save for the Nazgûl theme at AUJ's ending), yet here it's revealed in the least dramatic, matter-of-fact manner possible with no build-up or preamble, (say, such as Gandalf putting the clues together himself, speaking of which):
2. The Dol Guldur story is also a mess; how does one bury wraiths when by definition they have no bodies to bury? How does Gandalf leap to conclusion that Azog serves the Necromancer (because Beorn mentions an alliance between the Moria Orcs and the Necromancer? There are a lot of Moria orcs, and Azog is a "Gundabad Orc" anyway). If Sauron has not yet reached his full strength and is trying to remain concealed, why would he seemingly lure Gandalf/the Wise to Dol Guldur (I believe that was the implication of the "it's most certainly a trap" exchange); and why would Gandalf enter Dol Guldur alone if he even suspected that ME's darkest remaining power had returned (remember this is the same character who would not enter Moria unless he "had no other choice")? What happened to the Nazgûl after they were set up by the Morgul blade subplot? And would orcs really advertise the use of the fan-ficy "morgul arrows" and tag elf territory with Sauron grafiti if the Necromancer was trying to remain incognito? Also, why do the Orcs (other than Narzug), and Sauron only speak the Black Speech when they spoke largely Westron in LotR, and how does Thorin understand this language when Azog addresses him in AUJ (yet he has to be told by Gandy that the lettering in the prologue is Black Speech)?
3. How did the orcs wipe out a race of people who could transform into giant bears, as Beorn seems like he could fit Azog's entire head in his mouth? For that matter, why is Azog afraid of the last skin-changer when he's killed the rest, and why doesn't Beorn just slay Azog when he's right outside his house? BTW, how does Beorn still have a manacle on his arm when it would come off as soon as he turned into a bear?
4. Thorin and the other dwarves don't react when Kili is injured, yet totally overact most out of character when they fail to find the hidden door (no one mentions that they simply have to go back to Lake-town, and have the people dote on them until next year)? If Thorin and the others thought that map was false (as mentioned in Thorin's audition dialogue), this needed to be explained in the film:
"We were here...we were here when the last light of Durin's day fell, and there was no keyhole, no door...no way into the mountain. The map is a fake, a forgery, it must be...or else the hidden entrance was never built. But whatever the cause...the legacy of my people has lead to nothing". (Thorin throws away key).
5. Lake-town is a heavily populated/guarded police state, yet there's nobody around to notice a major brawl between orcs and elves in the middle of the night (after a thump in the armory brought a dozen armed guards to the scene at once)? It's also very convenient that the Master/Braga allow Bard to speak his piece before a large assembly, then decide to arrest him for no apparent reason (or none that they bother to give), at JUST the right moment for a climactic chase and to get Bard out of the way so the elves can handle the orcs themselves...and this is after they set up the same outcome with Bard being arrested for illegally trying to use the windlance, which IMO seems like the result of some sloppy last-minute re-shoots.
6. What is going on the the Arkenstone anyway? Smaug claims that this is what corrupted Thror's mind... yet Thorin begins to change before he reaches Erebor/the Arkenstone, and we're told by Old Bilbo and others that Thror's madness was brought on simply due to his lust for wealth...notice Thrain also went mad after exile from Erebor, after the Arkenstone had been left in Smaug's hoard. For that matter, why is the Ring already corrupting Bilbo when Hobbits are known to be resilient to its power, and Bilbo will hold the Ring with no obvious psychological effect for 60 years after?
7. No consistency with Film 1's ending - IE Thorin's magically healing face, and the orcs already being on the company's trail despite the eagles seemingly having flown them hundreds of miles thought the night and into the day. Notice everyone now calls Thorin "Oakenshield", which reminds me of a cheesy 80's action movie where our hero is always referred to by his last name. Tongue This also includes Smaug, who is exceptionally well-informed for someone who's been snoozing in Erebor for 60 years, but let's just chalk it up by his being telepathic to a degree as seems to be the case when he senses the Ring.
8. Two films in and aside from Thorin, Balin, and now Kili, none of the dwarves have emerged as distinct individuals with unique backstories or functions within the company. Yes, Oin's being the apothecary leads to his staying with Kili, and Nori a thief to his tapping for the hidden door...but these facts are not really established (aside from the supplementary materials) and also don't really add a whole lot more to the characters/movies; Dori and Ori, in whom we were supposed to be invested in the last movie, have pretty much ended up as extras. This isn't an "inconsistency", per say as a missed opportunity, IMO.
Those are just some of the issues I have with the films...I'm sure that they can all be explained away through various explanations of varying degrees of believability, except a good movie(s) shouldn't need to rely on the defenses of fans to make sense (such as LotR, which didn't seem to have nearly as many gaffs in its narrative construction).
. No doubt they'll do one for the Hobbit films. 
Isn't one of the most remarkable things about Tokien's work that he spent a lifetime trying to iron out and remove any inconsistencies - yes, I know he didn't manage it completely - but one of the joys of his work is that and that for every action there is a reason and a consequence.
