My Sister, the Homemaker - S5-E12
Visible crew/equipment: At the beginning of the episode, Jeannie is talking on the phone by the stairs and her sister is hiding behind the banister. Her sister stands up for a moment and shows her full face, revealing that it is actually Barbara Eden's stand-in wearing the green costume.
Wings Over Hooterville - S2-E1
Factual error: In a flashback scene, Oliver is hanging from a tree after getting shot down in WWII. They ask him where he is from, and he says the US Air Force. The US Air Force wasn't formed till 1947. During WWII, it was the Air Corps.
Other mistake: When Don and Dr Smith are stealing Sesmar's control unit, look closely at the base of the electronic arch - Raddion is hidden from sight but the careful eye will see his golden feet.