Character mistake: As Reddington sets up the blood transfer between himself and Ressler he inserts the intravenous line in Ressler's arm first and then in his own arm. This way the remaining air inside the line gets into the recipient's body and could very well kill him. Judging the expert way he handles the set up he should know to first insert the IV in his own arm, let the line fill with blood and then insert the IV in Ressler's arm.
Other mistake: After arresting the Reddington impostor, Ressler states that while hunting Red they had only one photo with a "passing resemblance to the man we've been working with." In S1E1 you can however see a high quality picture of Red on the front desk screen while he is arrested. (00:32:25)
Other mistake: Elizabeth picks up a stuffed bunny which supposedly was from her childhood. The bunny has burns on it after being in a fire. You can see 2 plastic tag fasteners attached to the bunny, which wouldn't have survived the fire.