When Sean is falling down the mountain on the ice, he loses his hat about halfway down, and there is no opportunity to retrieve it. It's still missing (correctly) as they go into to cabin and find the canoe. The next morning when they leave the cabin, he has the hat again - which he proceeds to lose again in the rapids. At least it didn't show up again.
Moments before the crash sequence of Jake's airplane, as he looks out the window, you can see the altimeter showing 8,380 feet in the foreground. The shot of the altimeter showing him looking directly at it shows around 2400 feet), moments later when apparently plummeting out of the sky, his altimeter is rapidly counting down from over 8000 feet again.
When Sean, His Dad and Jesse are eating dinner. Sean yells "what is wrong with you guys?" throwing down his fork. When his dad looks at him and we cut back to Sean the fork magically reappeared back in his hand.
When the kids are climbing down the icy slope there is a shot of Sean (just before Cubby shows up) where he holds the rope with both his hands In the next shot he's holding his axe instead.
The children are supposedly hiking up a river canyon toward a mountain called The Devil's Thumb. They stubble upon an old cabin. Inside they find and patch an old canoe, then proceed to head upriver where they encounter and are caught in rapids and a waterfall apparently flowing uphill.
At the end of Sean and Jesse's Rapid run in their canoe, Sean says "We did it" but his lips don't move.
Right after the helicopter comes to rescue the children and their father. The helicopter takes its turn around the mountain and you can see equipment and a head of a man in the left hand corner of the picture.
(at around 1h 08 mins) When the children are in the patched canoe, right after the bear floats on the drift log, you can see power lines stretching across the river in the far background behind the children, yet they are supposed to be in the remote wilderness.
Early on in the film the character of Jake Barnes gifts his children Jessie and Sean a figurine and a compass he says he picked up in Nome. However shortly thereafter Jake's small seaplane goes down near Devil's Thumb on a local fly run to deliver supplies to a nearby town. This implies they live on the southern Alaskan peninsula, more than 1,200 miles from Nome.
In the initial opening scene we see Jake Barnes flying his Super Cub south over glaciers and high rocky mountains, a feature only seen in the geography of South Alaska. When he checks in via radio, he confirms his location as being one in south-central Alaska. We then see him flying over a group of polar bears. Polar bears are only found along the northern coast of Alaska and Canada and on the Arctic Sea, not in the mountains of south Alaska (almost 1000 miles away).
When Sean is going down a rock wall to rescue his father from the plane, you can see the shadow of the microphone on a rock.