When Flynn is dying, he is on his back with his right hand palm up on the floor. Later when we see both lovers from far above (when the magic is working) his right hand is palm down. If Flynn is dead, he wouldn't be able to change the position of his hand.
During the first verse of the song "I have A Dream" (in the "Snugly Duckling"), the hook-hand thug hooks the keys out of the piano and they are sent flying across the room. In the next shot, he is continues to play the piano as if the keys were still there (which of course they can't be).
Rapunzel's hair is just long enough to pull Mother Gothel up, but it is also long enough to lower Rapunzel to the ground, which would take twice as much hair. However, it is seen that Rapunzel loops her hair over hooks when pulling Gothel up, which would allow for half the hair to still be in the tower, with just enough lowered to serve Gothel.
Rapunzel's "Mother" sees the light reflecting off the crown, and she knows that Rapunzel is gone. At the end of the movie, Rapunzel gives Flynn the crown. If she took it with her, then the mother would not have seen it in the tower. Mother Gothel gives the crown to Rapunzel to give to Flynn.
The goof item below may give away important plot points.
Incorrectly regarded as goofs
As "Mother" stabs Eugene after he climbs the tower to rescue Rapunzel, the knife comes out clean, no blood. However, blood on the knife would likely be reason to increase the MPAA rating to a PG-13, so to keep the rating at a PG, the blood was omitted.