At Ms. Smith's home, she mentions that the guest house was built to house escaped slaves during the Civil War. When the talking tube connecting the house to the guest house is shown close up, the end of the pipe is threaded. Threaded pipes were not in widespread use until much after the Civil War.
The locomotive leaves Philadelphia as a diesel electric. Later, in an overhead-view, the locomotive is electric, with a pantograph and other electrical equipment clearly visible. Later, it changes back to a diesel electric model.
(at around 11 mins) When the students and faculty are leaving the building, a girl in a blue shirt boards a bus. When Elliot and Julian are talking a little while later, the girl in the blue shirt is walking in the background.
(at around 4 mins) At the beginning of the film when the two girls are sitting on the park bench, the one unaffected girl is talking to her friend, but she doesn't respond. Claire, the unresponsive one, is looking down at her book, her eyelids nearly closed, but when the angle changes, she is looking straight ahead instead of down.
When Julian is in Princeton in the red Jeep, the Jeep is shown going down the street, not passing any driveways, and coming to a stop. In the next shot, the Jeep is shown stopped, having just passed a driveway on the right.
(at around 13 mins) When the news reporter is explaining the concept of a "preservation switch", the on-screen caption states that she is outside the CDC headquarters. The building in the background is actually the United Nations.
(at around 50 mins) After leaving Filbert, when everyone splits off into smaller groups in the field, Elliot goes to the truck to find a map. Only the passenger door is open, yet the door ajar alarm is dinging. It's a Ford truck, so the alarm would only ding if the drivers' side door was open.
(at around 8 mins) A science teacher would never say that something in nature is unexplainable. The entire point of science is to discover how things in nature and the universe work. Furthermore, a science teacher would never say that "Science will come up with some explanation, but in the end, it'll just be a theory." "Theory" does not mean "guess." Theories are supported by facts and logic. And his statement seems to imply that the truth just happens via committee or something.
(at around 45 mins) When Pvt. Auster is affected by the phenomenon and starts walking backwards, his foot slips but he controls himself to avoid falling which should not happen in people who lose the survival instinct.
Everyone left the diner at Filbert by car, but the majority came to town from the train that made an unexpected stop. Most of the people at the diner are either locals gathered with the others to receive information, or other visitors who came by car.
(at around 1h 12 mins) When Mrs. Jones bangs her head on the outside of the exterior wall, we see a picture frame bounce around as her head thumps the wall. This is impossible because that wall is stone - approximately 16 to 18 inches thick. The percussion of a human head striking on the exterior cannot transmit to the interior.
(at around 1h 7 mins) After dinner, Ms. Jones tells the group that the guest room is up the stairs and to the left. Given the position of the stairs to the room when Elliot talks to Ms. Jones from the top of the stairs, the room was to the right.
The goof items below may give away important plot points.
(at around 33 mins) Julian is sitting in the front passenger seat of the Jeep and is not wearing a seat belt. When the driver speeds into the tree, two people fly out of the front windshield--neither of which are Julian. Not only is this preposterous, given that Julian didn't have a seat belt on, but this also means that someone behind him flew through him to fly out of the windshield.
(at around 24 mins) When the cell phone video shows a man getting mauled at the zoo, the phone and a woman's fingers are perfectly still. Later, the same women has the same phone from a different angle, and she is constantly shaking.
The plants are releasing a neurotoxin that turns the brain's self-preservation mechanism into a self-destruct mechanism. When Mrs. Jones is affected, she rams her head into the windows to let the gas into the house. That could be a more efficient way to self-destruct. Similarly, the driver of the car Julian was in tried to kill himself instead of others inside the car.
(at around 22 mins) When the train stops in Filbert, the conductor tells Elliott that they've "lost contact." When Elliott asks who they lost contact with, the conductor replies, "Everyone." People get off the train and use cell phones, and the stranded passengers watch TV news in the diner (which has power). The train engineer is probably referring to the train controllers, not everyone in the world. He is likely also exaggerating due to his emotional state.