When searching Rudy's apartment, Dexter goes over to look at the Barbie record player. The African-American police employee who is cataloging the items is clearly wearing a laminate with a photo of a bearded, white, middle-aged man on it.
Dexter finds the story of Laura's murder in the October 3rd edition of the newspaper. The October 3rd paper would report on what happened October 2nd. Brian would have known that they were actually "born free" on 10/2, not 10/3 (leaving aside the fact that if they were found on 10/2, then the murders actually occurred on 9/30).
The goof items below may give away important plot points.
Crew or equipment visible
At the end of the episode, when Dexter flips the switch sending Rudy Cooper/Brian Moser upside down, the tube pumping blood out of his neck is clearly visible to the left running up the metal frame and out the back right side. Even a piece of black gaffer tape is seen in the second/final shot stuck to Rudy Cooper/Brian Moser's chin.
When the Police finds the body of Bryan and they are analyzing it, in two scenes the head of Bryan who is up-side down is shown and you can clearly see the throat of the corpse (Bryan) is moving, that is he is breathing.