Larry Hagman mentions this as one of his most memorable episodes because of the scene with the elephant. The "magic" on the set was done by the actors "freezing" in place while whatever Jeannie blinked is brought onto the set and then filming is resumed. So as Larry was frozen in place and they navigated the elephant onto the set (a very tight fit), the elephant suddenly "broke wind" right in Larry's face.
When Jeannie is all dressed up and Dr. Bellows shows up at the front door, she disappears behind the door using a movable panel in the set, as she did in How Lucky Can You Get? (1966). There is no cut; the disappearance happens in real time.