According to George R R Martin:

A Song of Ice and Fire is told through the eyes of characters who are sometimes hundreds or even thousands of miles apart from one another. Some chapters cover a day, some only an hour; others might span a fortnight, a month, half a year. With such a structure, the narrative cannot be strictly sequential; sometimes important things are happening simultaneously, a thousand leagues apart.

Which chapter in the series covers the longest amount of time?

Bran III, A Dance with Dragons

Length: Seven weeks (a full moon's turn, plus three-quarters of another)

The following are exerpts from this chapter. On the first page:

The moon was fat and full. Stars wheeled across a black sky. Rain fell and froze, and tree limbs snapped from the weight of the ice. Bran and Meera made up names for those who sang the song of earth: Ash and Leaf and Scales, Black Knife and Snowylocks and Coals. Their true names were too long for human tongues, said Leaf. Only she could speak the Common Tongue, so what the others thought of their new names Bran never learned.

Later in the same chapter:

The moon was a black hole in the sky. Wolves howled in the wood, sniffing through the snowdrifts after dead things. A murder of ravens erupted from the hillside, screaming their sharp cries, black wings beating above a white world. A red sun rose and set and rose again, painting the snows in shades of rose and pink. Under the hill, Jojen brooded, Meera fretted, and Hodor wandered through dark tunnels with a sword in his right hand and a torch in his left. Or was it Bran wandering?

Later:

The moon was fat and full. Summer prowled through the silent woods, a long grey shadow that grew more gaunt with every hunt, for living game could not be found. The ward upon the cave mouth still held; the dead men could not enter. The snows had buried most of them again, but they were still there, hidden, frozen, waiting. Other dead things came to join them, things that had once been men and women, even children. Dead ravens sat on bare brown branches, wings crusted with ice. A snow bear crashed through the brush, huge and skeletal, half its head sloughed away to reveal the skull beneath. Summer and his pack fell upon it and tore it into pieces. Afterward they gorged, though the meat was rotted and half-frozen, and moved even as they ate it.

Later:

The moon was a black hole in the sky. Outside the cave the world went on. Outside the cave the sun rose and set, the moon turned, the cold winds howled. Under the hill, Jojen Reed grew ever more sullen and solitary, to his sister's distress. She would often sit with Bran beside their little fire, talking of everything and nothing, petting Summer where he slept between them, whilst her brother wandered the caverns by himself. Jojen had even taken to climbing up to the cave's mouth when the day was bright. He would stand there for hours, looking out over the forest, wrapped in furs yet shivering all the same.

Later:

The moon was a crescent, thin and sharp as the blade of a knife. The days marched past, one after the other, each shorter than the one before. The nights grew longer. No sunlight ever reached the caves beneath the hill. No moonlight ever touched those stony halls. Even the stars were strangers there. Those things belonged to the world above, where time ran in its iron circles, day to night to day to night to day.

(The intervening quarter-moons are mentioned as well, but they aren't needed here.)