The birth year for Thorin given on the tapestry that Bard finds in Lake-town is correct for the book's continuity. However, in the film-trilogy, Balin is supposed to be the oldest member of the Company, Thorin allegedly being several years younger than Dwalin.
When Gandalf fights Azog at Dol Guldur, his sword falls from his hand and he is thrown backwards. Immediately afterwards, his sword is sheathed at his waist, despite the fact that he is now holding his staff in both hands and could not have sheathed it.
When Azog calls for Bolg in Dol Guldur there is a side shot of the two approaching each other where Bolg is clearly taller than Azog. When they are speaking in the close-up Azog is looking down at Bolg implying that he is taller.
When Bilbo looks to the treetop in Mirkwood, he can see the sun-rays through the branches as if the sun was in the middle of the sky. It implies that it's midday, but when he climbs a tree to watch above the forest, the sun appears in the horizon, as a sunset.
It was established in The Hobbit: An Unexpected Journey that Thrain is missing his left eye (This has been confirmed in several official Hobbit publications). Yet in this film he is shown with both eyes fully intact.
When Tauriel is searching the cells for the missing dwarves she is wearing one outfit with her breastplate armor. Seconds later when she arrives in the wine cellar she is wearing a completely different dress with a belt.
At the furnaces, when Thorin taunt Smaug into breathing fire to start up the furnaces, there are plenty of wooden objects (benches, crates,...) in the path of Smaug's fire breath, that will not burn, yet the coals in the furnace almost instantly catch fire.
Beorn still wears a shackle on his wrist from his time of enslavement by Azog. However, the shackle is too small to fit his arm in bear form, nor was it present when he was a bear the previous day. He should have already broken free of it.
The movies previously retained the idea from the books, that sunlight renders Orcs non-functional. Gandalf remarked in The Lord of the Rings: The Fellowship of the Ring on how alarming it is that Saruman's breeding program has produced sun-resistant Orc soldiers. Yet in the "barrel" escape-scene from Thranduil's caverns, the orcs are chasing the dwarves down the River Running, and doing hard battle, in broad daylight without suffering any ill effects.
In the treasure vaults of Erebor, Smaug tells Bilbo that he'd almost be willing to give the Arkenstone to "Thorin Oakenshield" just to watch it drive him mad. Smaug has no particular reason to know the name of the grandson of the king he deposed from Erebor, and since he's been sealed inside the lonely mountain ever since, he certainly wouldn't know the "Oakenshield" name that Thorin earned in a battle at Moria years later.
When Bilbo puts on the ring in the forest, he can hear what the spiders are saying, in English. He takes off the ring and stabs the spider with his sword, at which point, we can hear the spider cry, "It stings! It stings!" although the ring is no longer on, and Bilbo should no longer have the power to understand the spiders' language.
Just after Thorin rides down the river of gold on a wheelbarrow, Bilbo is forced to leap off of the falling bridge knocked over by Smaug. Upon hitting the ground, he rolls and Sting at his side can clearly be seen to bend almost at right angles, indicating a rubber stunt sword.
As the company prepares to enter Mirkwood, the skies become overcast and it begins to rain. During this scene, there are shots of Gandalf where light and shadows show that he is standing in bright sunlight instead of overcast skies, with digital rain added in after filming.
The goof items below may give away important plot points.
The number of barrels changes from the time they leave the cave to their arrival in Laketown. Bilbo counts 13 barrels going down the trapdoor; he clings to one; Bombur leaves a broken one among the orcs before jumping into an empty barrel; Bard loads 14 barrels on his barge and later has them filled with fish.
It would be impossible for a barrel to constantly float right side up when it is going over rapids, constantly tipping over and unavoidably filling with water. Also, physics denote that an opened barrel with a heavy object in the bottom (i. e., a dwarf or hobbit) would immediately want to tip over on its side and float that way, as opposed to floating right side up the entire time the company is in the water.
When the dwarfes use the gold in the furnaces to trap Smaug, the gold is melting way too fast. The heat necessary would either melt the stone of the furnaces as well or the thermal stress would crack and break them long before the gold to be heated enough to liquify.