After the female robot bunny falls apart all the pieces that fell to the floor are gone in the next shot.
When Bugs announces that he is going to "...exit, stage right.", he is actually moving stage left; "stage right" and "stage left" are opposite the directions as seen by the audience. Bugs should have said either "stage left" or "house right".
When Bugs is being held by the neck by Gossamer/Rudolph, he says, "Look, out there in the audience." When he says the "Out there in the audience" part, his mouth doesn't move. The Hair Raising Hare Schematics on the Looney Tunes Golden Collection prove this: during the part where you see the goof, you also see that the "out there in the audience" part was added in a different color ink.