During the opening montage of the various pets acting up while their owners our out, Mel is shown barking at a squirrel that is in a tree outside his window. At the end of the movie as the camera zooms away from the building, Mel is shown jumping around one floor below Max's apartment, which is 7-8 stories up. There is no tree outside of Mel's window (or any tree at all).
When Max is waiting for Katie to return and sitting in front of the door near the beginning of the movie, you can see a coat rack on the wall to the left of the door with a blank wall above it. Later, the guinea pig Norman comes through an air vent that has miraculously appeared above the coat rack.
During Duke's first night in the apartment, Max gets into the doggie bed and scoots all the way to the far side of where Duke is sleeping on the floor. When the camera cuts, Max is in the middle of the doggie bed.
The small table near the front door in Katie's apartment keeps moving. First, it is right beside the front door with the air vent to the right of it. Later, it moves around the corner to beside the bedroom door, with the air vent to the left of it. When Katie returns near the end, it's back to being right beside the front door.
When Duke drags Max from the park by his lead we see Max being pulled through tight railings in an apartment block staircase. For Max and his lead to be pulled between the railings, Duke must have passed through first. Impossible since Duke is several times bigger than Max.
when the truck falls into the river it is broad daylight. When the scene cuts to Snowball diving in to save Max and Duke from the truck, it is suddenly night time. The rapidly-sinking truck could not have been in the water for more than a couple of minutes, so this is impossible.
When Max is trying to save Duke in the end, he is walking on the windshield of the truck to get to the keys. Both sides spider and shatter underneath him. Ford started requiring all their vehicles to have laminated safety glass in 1929. It is now a standard for all vehicle windshields. This would not have splintered and shattered away underneath the dogs feet. It would have been relatively safe for him to walk across it with no issues.
After Snowball throws down the keys and Max opens the cage they swim right out of the van without opening the doors. However we see the windshield shatter, thus making a clear way for the three to swim out.
The first time Max and Duke got into the water they could barely save themselves by swimming to a boat a few meters away. The scene makes clear that Max cannot swim. But in the end Max saves Duke by swimming to the sinking truck and then they even swim nearly 100 meters to the end of the bridge.