Deliberate mistake: When the Darlings discover the dead rat, they raise the curtain and cry out loud, but the rat is nowhere to be seen. Given Disney's long time history of refusals to show dead bodies or blood, it's highly possible that this was a deliberate mistake, but really awkward indeed.
Continuity mistake: When Lady is running away from the pet shop she is chased by the angry mutts. The middle dog's inner ear color is solid green yet in the next scene is pink. (2006 edition).
Tony: Hey, Joe! Look! Butch-a, he's got a new girlfriend.
Joe: Well, a-son of a gun! He's a got a cockerel Spanish-a girl.
Tony: Hey, she's pretty sweet kiddo, Butch. You take-a Tony's advice and settle down with this-a one, eh? Hehehe.
Lady: "This-a one"?
Tramp: This-a one... this-a... Oh! Tony, you know. He's-a not-a speak-a English-a pretty good.