Continuity mistake: When the children are singing, "So Long, Farewell", as they are going upstairs they pass a picture of Lois' parents hanging along the middle of the stairwell. When they get to the top of the staircase, we see the exact same picture, but now at the head of the stairs.
Continuity mistake: In the part where Meg is asking Brian about God, and Brian answers he is an atheist, the camera pans out quickly, and we see an entrance to the kitchen with an air vent above it. Eventually the family discussion with Brian continues, and you see that the air vent is now gone.
Plot hole: When it is revealed that Chris's teacher was never born, Teacher Doug introduces himself, despite the fact that if Chris's teacher was never born, Doug would already have been teacher for the entire year, and therefore all of the students should know him and he would not have to introduce himself.
Continuity mistake: When the crowd is outside Peter's house wanting Peter to heal them, a lady in pink comes out and says "It's him!". While she does this she is standing on the lawn, but in the next shot she is standing on the cement path. Other people in the crowd change position in this cut too.
Continuity mistake: Before Dr. Hartman delivers Bonnie's baby, he removes a small wheelchair and places it on a cabinet next to him. After delivering the baby, he makes a funny remark about Paula Poundstone, and then he hands the baby to Bonnie. Notice that the wheelchair and the cabinet are no longer present.
Other mistake: Peter goes back in time and sees Lois at age 18. Death says everyone will see Peter as being 18 too. Problem: Lois and Peter aren't the same age. In later episodes, Lois reveals she's 2 years older than Peter so Peter should be 16 when he meets Lois.
Continuity mistake: When we first see Kate and her parents at the restaurant there are only 2 empty chairs. Stewie then sits in the chair next to Kate. After Kate's parents ask what happened, we see a chair in between Stewie and Kate, but you can also see the edge of third empty chair. A plate and the wine filled glass also move.