This is hinted at in the theatrical release and made more explicit in the extended edition: Thranduil wanted the white gems, the Dwarves wouldn't give them to him, so one of the reasons why he doesn't help is out of spite.
The idea here is to weave in elements of the ancient quarrel between the Elves and the Dwarves. Jackson, of course, is not legally allowed use the Silmarillion material, so he can't make explicit references to the Ruin of Doriath or the slaying of Thingol, so this is his way of dealing with the matter. (Have I tempted you to read the books yet?)
If an Elf acting out of spite doesn't seem in keeping with the "woodland hippy" cliche that's embedded in popular consciousness: it's very much in keeping with some of the Elves of the First Age, who could be a fairly hot-headed squabbling bad-tempered lot (Thranduil is written this way in the second Hobbit movie).
It's probably good to throw in a quote from the Silmarillion here:
'How do ye of uncouth race dare to demand aught of me, Elu Thingol, Lord of Beleriand, whose life began by the waters of Cuivienen years uncounted ere the fathers of the stunted people awoke?' And standing tall and proud among them he bade them with shameful words be gone unrequited out of Doriath.
It's also a practical matter. As Bilbo says in his introduction:
For this city lay before the doors of the greatest kingdom in Middle-earth: Erebor. Stronghold of Thror, King under the Mountain, mightiest of the dwarf lords.
If a dragon has just successfully overthrown "the greatest kingdom in Middle-earth" then Thranduil and his people really don't stand much chance against it, especially not in open warfare. Again, from Bilbo's introduction:
Thranduil would not risk the lives of his kin against the wrath of the dragon.
So there you have it: greed, spite and caution.