Jack's taxi is quite visibly battered with its side view mirrors broken off after he and the kids first elude the government agents. In the next (long view) shot, the mirrors are intact. A moment later (and in a close up shot), they are again broken.
When Jack and the kids sit down at the table in the restaurant the ketchup bottle is empty. but as they are talking the ketchup bottle is partly full, and has ketchup up the sides and top.
After Seth stops the government vehicle the knocked out driver's sunglasses are hanging from his his forehead. While Jack and Burke exchange words, the driver is coming to and the glasses are still on. Right before Jack walks away, the driver's head is leaned back in the seat and the glasses are nowhere in sight.
When the RV is shown leaving Las Vegas behind, the sun is on the horizon which would indicate sunset or sunrise. However, in every scene prior or afterwards, the sun is high in the sky.
At the table scene in the restaurant, during a short conversation between Jack and the kids, food and drinks appear on the table, without being served.
When Sara levitates Dr. Alex Friedman's MacBook at the convention the keys appear to have an under glow, but only Apple MacBook Pro's have this feature.
When the kids are captured their street clothes are replaced with white scrubs and they are restrained on the operating tables. Later, when Bruno and Friedman rescues them the kids have their hand held alien tech with them, having them rather prominently on display in the lab, so they could have grabbed it on their way out.
When "Jack Bruno"' is driving his taxi to the small country town, at least the two right side wheels are seriously out of alignment and both the front and rear bumpers are damaged. When they leave the town the Taxi wheel alinement's is perfect. Considering the garage mechanic only had one hour to do the repairs and it is very highly unlikely he would have color-coded bumpers for an out-of-town taxi, he would not have had enough time to complete the repairs as listed by Jack Bruno let alone the other repairs as seen as they leave town. However, there is nothing that suggests the mechanic didn't re-align the wheels first thing. It could have been that he started with the most obvious first and then moved to under the hood.
When talking about the Mustang on Jack's key chain which is hooked to the key in the ignition in the RV. The RV isn't Jack's, so it shouldn't even have been used at that time. However, given the sentimental meaning behind the key chain (as discussed in film) there's no reason he couldn't have removed it from his own key chain before handing the keys to the cab over.
When Sara and Seth are captured outside of Witch Mountain, the "mountain that doesn't exist", you can clearly see traces of chalk powder used by rock climbers all over the "wall" behind them.
When Jack and Dr. Alex gain control of the truck that is taking them away from Witch Mountain, the gear selector is clearly in the "Park" position even though they are "driving" (most obvious when Jack is in the driver's seat).
The opening credits mix newspapers and live action UFO footage. One of the newspapers reports the Roswell incident and uses a date of 1958. However, the Roswell incident took place in 1947.
When the RV is first seen outside of Las Vegas, the camera moves off the road and rotates to follow the RV. When it rotates far enough, the RV drives through dust where the camera was originally. The dust could only have been disturbed by a helicopter's rotor wash.
When the agents are initially searching for the kids, and they see them get into Jack's cab the lead agent, Burke, requests "Synchronous satellite feed! Get me a license plate." even though the cab number (1975) is clearly visible on the bumper. With that cab number they wouldn't NEED a license plate, as they'd have the cabs low-jack location data and radio frequencies, and could track it in real-time from the cab company's office (as the dispatcher did when he called Jack on his way out of town).
Dr. Alex Friedman said to Jack Bruno that "closed-minded skeptics are always welcome". This was the wrong use of the word. Within the scientific circles from which Dr. Friedman comes, "skeptic" refers to somebody who rigorously uses the scientific method (the opposite would be "pseudoscientist"). Skeptics are, by definition, open-minded and Dr. Friedman herself is actually a skeptic, so that is most likely the word that she meant.