Continuity mistake: When Bugs is viewing footage of the previous encounter between Cecil Turtle and himself at the start of the cartoon, you can see that viewed from behind, the projector only has one reel. When we see Bugs eating his carrot in the next shot, there are now two reels.
Continuity mistake: When Cecil Turtle is speaking to Bugs (in disguise) outside his home, he uses Bugs' false beard to dry the plate he is holding. When we see the two talking later on, the plate is nowhere to be seen.
Continuity mistake: When Bugs sees the finish line, you can see there is nothing on the road and nothing by the finish post. But when Bugs approaches the finish, there is now a line on the road and the post is in front of a tree.