Around 01:26:47, it's twilight. Around 01:27:22, it's daylight.
Around 01:25:02, Elliott lands at around 100 meters in front of the pickup. Two shots later, he's around 20 meters of the pickup.
When Pete first meet's Natalie in the woods he says he's 5 when he should have said 4. Pete could possibly sure of his age so he must have guessed 5.
When Grace marks the tree with blue spray paint and mentions that the owl and the tree will be safe now, she is actually marking the tree to be cut down. The U.S. Forest Service works with logging companies to thin forests by marking certain trees with blue spray paint so the loggers know to cut them down.
Losing air pressure on the brakes would cause the truck to stop, not losing its ability to brake in total.
When Gavin tells Jack about his first encounter with Elliot, Jack doesn't believe him at first and then suggests to call the forestry in the morning. Gavin fails to mention the three other eyewitnesses as well as the pickup, which would have certainly been damaged when Elliot stopped it, thus proving his story. However, as we learn later, his goal was to capture the dragon and take credit for it, so he needed help from someone who he could trust. Realizing that Jack isn't interested in his goal (and feeling hurt about it), Gavin walked away in anger, not even wanting to convince Jack of anything anymore.
Grace tells Pete she was "not much older" than him when her mother died. Later in the film, her father says that Grace was "maybe 5 or 6, just before [her] mother died", whereas Pete is about 11. However, Grace actually meant that she lost her mother at about the same age Pete lost his parents. Pete was 5 when that happened (and by that point Grace already knew that), so she was correct.
Pete and Natalie falling from the tree would have realistically resulted in them getting cut and dirty.
Several vehicles are too new to be in 1983. The Park Ranger Jeep Cherokee driven by Grace was introduced in late 1983 but the model in the movie has rear shoulder seat belts which would make it a 1990 model or later.
There is a scene when they enter the main street and it is set up like an American street, i.e.: drivers on the right side and marking on the right.. BUT of on a side street, they have failed to correct the New Zealand road markings and cars are on the left.
Karl Urban's New Zealand accent leaks through when Gavin remarks that Elliot's tracks are "less than a day old."
Near the end, Meacham (Robert Redford) can be heard saying that "the North star shines brightest of all". This is a common misconception. In reality, the North star is only about 50th brightest (its exact placement is impossible, because it's slightly variable).