Continuity mistake: When the female owner of Beethoven's girlfriend (Virginia or something like that) tells her boyfriend (Floyd) to put the leash on the dog, he goes to do so but when you see his hand clip the leash on, the shot changes to a woman's hand clipping the leash on (the nails are covered in nail polish).
Revealing mistake: When Ryce is talking to the vet on the phone, the vet would have to be talking extremely fast because of how little time there is between Ryce's questions.
Continuity mistake: In the opening of the movie, the little girl asks if they are late. When the family gets into the cars, she's not there.
Other mistake: The puppies constantly change age/size between scenes. Notably near the beginning, in the basement they are old enough to be walking around when the janitor is whistling to them, and a few minutes later in the Newton's basement they are nearly newborn. When the family go on holiday, when Ted is showing the puppies to the girl with braces, they have grown quite big to nearly adult size, and in the next scene where Mr. Newton is chopping firewood and they are cooped up, they are pretty young again.