When Aladdin and Jasmine make up in the palace garden (during Iago's song "Forget About Love"), Aladdin is wearing his "Ali Ababwa" prince clothes. When the we return to them after a scene with Jafar and Abis Mal, Aladdin and Jasmine are still in the palace garden, but Aladdin is suddenly wearing his "streetrat" clothes.
When Aladdin gives jewel rose to Jasmine, Jasmine puts rose on dressing table and tells Aladdin that sultan is going to make a royal announcement, the rose keeps appearing and disappearing in background.
When Aladdin, Jasmine, and Genie are singing "Quite Like a Friend", at the end we see Aladdin on Genie's right side and Jasmine on the left, but in the shot that immediately follows, it appears that Jasmine and Aladdin have switched places.
When Aladdin is led to his execution, his wrists are chained in front of him. This continues all the way to when he is kneeling over the chopping block. When Genie swoops in to save him, his hands are now tied behind his back.
In the opening sequence, when Abu unloads the treasure off of the carpet, Aladdin says something to the effect of, "That's the last of it, Abu". In the subsequent shot, a bag of gold can be seen behind Abu.
During the dinner with the Sultan, the Genie cuts off his hand and it starts to tap dance. When it stops dancing, the Sultan's eyebrows are gone. Then when the Sultan starts laughing, his eyebrows return.
When Aladdin takes Iago to the palace to get him a fair trail, Aladdin locks the cage he puts Iago into with a key. Later on, when Iago is shouting at Abu to open the cage, for him, the cage now no longer has the lock on it from earlier, enabling Abu to open the cage for him.
During the song 'Nothing In The World', Genie tells Aladdin and Jasmine about his travels around the world. He mentions being at the pyramids at Giza in Egypt, but then in the same part the three are shown at the Temple of Ramesses II at Abu Simbel with the pyramids in the background, even though Giza is in the north of Egypt, while Abu Simbel is almost as far south as it's possible to go in Egypt. However, what we see is clearly not an actual location, but Genie's "virtual reality" where anything is possible. After all, the statues at Abu Simbel have Aladdin's and Jasmine's faces, indicating that it's all not exactly real.
The goof item below may give away important plot points.
Incorrectly regarded as goofs
Throughout the film, Genie is seen wearing his golden shackles, even though he was released from these shackles at the end of "Aladdin" when Aladdin wished him free. This is not an error, but a design decision (see trivia). Within the concept of the movie, the Genie must have conjured himself a new pair of shackles, possibly because he was so used wearing them for many thousands of years. In any case, given his powers, any discontinuity in Genie's appearance is explainable.
After Genie saves Aladdin, he frees Carpet, Jasmine, Abu, and the Sultan. When he gets all four onto a roof of one of the palace towers, the Sultan's turban has reappeared after Jafar stole and slashed it earlier. Apparently, Genie restores it with his powers when he frees the others.