At breakfast, when it's Jo-Jo's turn with his dad, 12 seconds start on the clock, but when the mayor starts showing Jo-Jo all the perks about being mayor, the clock goes back to 12 seconds left when it should be at 5 seconds left.
When we first see Jo-Jo, the narrator introduced him as "the smallest Who of all." But seconds later one of his little sisters appears in the same shot and she is clearly smaller than he is. This is because the sister is a baby and therefore supposed to be smaller then Jo-Jo. The term "smallest Who of all" could be a reference to him being small then other Whos his own age (as shown in a deleted scene).
The mayor explains to his son that the family has been in Whoville for centuries. A few scenes later we learn that the city is planning their centennial. Actually they are celebrating their one hundred years of happiness and peace, so his family could have been there for centuries.
In order to save the speck of dust, Horton quickly sucks up some water from the pond and swallows it. However, elephants can't drink through their noses. They use them to squirt water into their mouths when they're thirsty. Elephants also can't climb trees, stand upright, hold objects with their front feet, move their ears like hands, or talk. However, Horton is a Dr. Seuss-style elephant from a fictional world, so all these stunts are feasible.
When the mayor is telling Horton about his family, he says "I have a beautiful wife, ninety-six daughters and one son." He doesn't mention his son's name. Yet later in the movie, Horton says "The mayor has one son named Jo-Jo." Unless he and the mayor have another conversation in the movie that we don't see, Horton couldn't have known Jo-Jo's name.