Carl uses sails to propel and steer his house. While sails don't work if the balloon/house are moving at the same speed as the air mass, they do work whenever the wind changes speed or direction. The house has significant mass (weight) and so does not respond instantaneously to wind changes.
When the house first rises, breaking free of its foundation, the front porch is clearly empty. Yet Russell appears later in the far right corner. This is explained by Russell later when he says he crawled under the porch chasing a "snipe". Apparently he hung on to something underneath the porch and crawled up later. In the short film George & A.J. it is revealed that Russell braced himself underneath Carl's house when it took off in the same way that Woody held onto Sid's milk crate when Sid picked it up in Toy Story. However, in the Up movie several shots show the complete underside of the house as it flies off and Russell is clearly not there (or on the porch).
When they are flying and Russell speaks of the cumulonimbus cloud, Frederickson turns his hearing aid off by turning the knob forward. When the lightning strikes, he turns it back on... by turning the knob forward.
In the shot where young Carl is running with the balloon and stops near a house listening to a sound. His shadow is at his right side and after he turn to his right the shadow is in front, but the next shot shows the shadow falls behind him not in front.
In the French version of the film (with possibly analogous errors in other language versions), during the scene at the beginning when Young Ellie is showing Young Carl her scrapbook, a close-up of one of the pages says in French "TRUCS A FAIRE" but when other side shots are shown, the book clearly says in English "THINGS TO DO".
In the scene where Carl and Ellie paint the mailbox it can be seen that Carl's hand spoils the paint with his hands paint making a mark on the mailbox to which Ellie reciprocates and make a mark of her hand. In the later scenes its found that mark made by Ellie is rotated clockwise.
After Russell frees himself from being tied up, he was seen standing on the hose nozzle. However, he was then found clinging onto the nozzle when he was dragged across the front of the airship. Right after that, a far view of Russell showed him standing on the nozzle again.
In the scene where Russell finds the lever to operate the house direction he moved it first towards right then to its left and again to its right and finally leaves it in the center and goes towards the window, but in the next shot its found positioning to its right.
In the shot where Carl notices his mail has arrived, he holds his cane in his left hand and reaches the mailbox. He places the cane to his left; in the next shot he picks up the mail and comes back, and the cane is not visible. Then in the shot afterward it is found to Carl's right side, when he speaks to the man in the black suit.
The phrase "Jiminy Cricket" was used in common slang as a euphemism for "Jesus Christ" (in context of a "cuss" word) since the 1920s. The expression is spoken in movies like The Brave Little Tailor and The Wizard of Oz. Pinocchio had a character named Jiminy Cricket who was named after the expression, not the other way around.
Carl's house starts flying at the moment when he releases the balloons from the chimney; assuming that all balloons were already inflated and stored inside the house, this should have made the house float hours before already, regardless of whether they are inside or outside the house. However, it appears the balloons were in a giant sack in the backyard, and could have either been tethered down securely, or compressed to reduce lift.
Soon after the house is first launched and has clipped two TV aerials, an external shot of the house roof shows the weather vane rotating so the arrow is pointing south. This indicates the wind is blowing from south to north, and could not possibly take the house to South America.
When Carl and Russell first stop the house from drifting off without them, Russell does not have his backpack and Carl does not have his cane. However, right before the house hits the ground, while they are waiting on the porch, Russell is wearing his backpack and Carl has his cane. Everything is scattered from the impact and Carl and Russell chase the house and stop it right before the cliff. They simply backtracked to reclaim their belongings.
The age timelines of the characters don't add up. When we see Carl as a kid, he is maybe 10 or 12 years old and Charles Muntz appears to be in his late 20s. Throughout the rest of the movie, Carl appears to be in his 70s. This would mean that Charles Muntz would be around 100 years old. This would make some things that he does highly unlikely for his apparent age.
In the title scene where young Carl Fredricksen is running with his balloon in the hand and smashes at the tree. The tree's shadow is missing behind and only young Carl has his shadow, as if he's floating in air.
The goof items below may give away important plot points.
Incorrectly regarded as goofs
Carl's cane always has four tennis balls on it even after he throws one or more of them to the dogs in multiple scenes. When Carl throws the ball to Dug, Dug brings the ball back to him. However, when he throws it at the pack of dogs later on, he doesn't get it returned. Hence the reason why he only has three tennis balls in the following scenes.
Charles Muntz's airship is on its way back to U.S. when Carl's house is following him, and during the struggle to retrieve Kevin the rare bird. After all this apparent flying on the same direction, Carl's house falls at same spot near the falls. However, it slides off the wing of the airship, and so becomes at the mercy of the wind.