When Hank is navigating Dory with the stroller through the park, they knock down a little girl. Before she is struck, the girl's shirt is blue-and-white striped. When she is seen a moment later, crying on the ground, her shirt has changed to solid pink.
Even though this movie is set one year after the events of Finding Nemo (2003), Nemo is shown to be the same age as he was in the first film. Over the course of a year he should have grown from being a juvenile to at least being an adolescent, if not an adult.
When Dory and Hank meet for the first time, Hank pulls the cart she's on twice towards him to be positioned in the same spot each time.
When Dory is recalling the moment in which her parents explain to her little self about following shells, sand that coves up one of the shells partially disappears completely when the camera follows grown up Dory.
When Nemo approaches the sea lions he is in front from the rear angle. From the front, he is behind his dad.
Bailey the beluga whale and Destiny the whale shark are kept in an adjoining enclosure, separated by a rock wall with a metal grate that indicates they are sharing the same water. A beluga whale makes its home in Arctic and sub-Arctic waters, whereas the whale shark is indigenous to tropical and sub-tropical waters. Factually, they would not be kept adjacent to each other in shared waters.
Becky is a common loon (a.k.a. a great northern diver). The legs of loons are set so far back that they are unable to walk or even stand up on land/buildings, as Becky is shown doing, and in the rare cases where they mistakenly land on the ground are often stranded with no way to take off again. As a result of their reliance on water, they eat only aquatic animals, and would not likely stop to risk their life to strand themselves on land to eat some popcorn.
While octopodes have been observed crawling across land between tide pools, they do need water to breathe and are not nearly as amphibious or agile as Hank is portrayed.
When Hank is found in the coffee pot looking like a potted plant, he would have displaced all of the water, yet when he climbs back out the pot is once again filled with water.
Hank is shown to have permanently lost a tentacle, whereas real octopus can regrow any arms that have been severed.
Dory's scars from her travel through the jellyfish in Finding Nemo (2003) have disappeared. They may well have healed in the time between the two films.
Dory tilts the coffee pot (to be dropped into the bucket of dead fish) by moving to the pot's left side. But fish maintain neutral buoyancy, so moving inside the pot would not change its center of gravity. She didn't merely shift its balance, but nudged the side of it.
Although Destiny states that she's a whale, whale sharks are actually fish, the largest fish in the sea, as evidenced by how her fins are positioned and that she lacks a blowhole. Though her tank is presented in a way to support marine mammals like dolphins and smaller whales.
When the fish going to Cleveland are in the van there should be lids on the tanks. This would mean that the fish wouldn't flop between tanks and Dory and Hank wouldn't be able to get to the sun roof.