I've been spending some time at a remote resort recently with little for company except a DVD set of I Dream of Jeannie. Why is it that Major Nelson has to keep Jeannie a secret from NASA and Col. Bellows, M.D? She is practically all-powerful, so it's not like she could be exploited by them, and you would think that people with such a dangerous mission would appreciate having a genie up their sleeves.
Let's see - an astronaut has a genie that can go back and forth in time and all over the world and even in space.
Can you say, "Security risk?"
So you have that, along with two other issues:
1) Security risk
2) Threat (she can stop any weapons the US has and cause as much damage as she wants)
3) Potential weapon - if they could study her and understand why she has powers, they could harness them (or so would many think) so she'd become a lab rat.
As a non-human from another plane of existence, she's not actually a US citizen. She's actually an illegal alien.
Keep in mind: The show is set in the 1960's. And pretty much adheres to 1960's paranoia and social standards.
Her association with Major Nelson is a security risk in the (rather acutely paranoid) 1960's. Were it discovered that she was a non-human (as evidenced by her red and green blood), it would create a massive paranoid theory inspired sweep of all persons associated with government agencies.
Further, it would reveal multiple criminal actions by Maj. Nelson. At the very least, Major Nelson would be expelled from NASA. At worst, he'd be bound over for multiple national security violations, including foreign travel without authroization while on duty, multiple counts of theft, slavery, and unlawful cohabitation. It could be argued that he committed treason, even, though I don't personally think that would stick even as far as trial.
Further still, Major Healy would also be implicated as an accessory in many of the above mentioned crimes.
So, for fundamentally selfish reasons, Major Nelson needs to avoid it being discovered he's keeping an alien bond-slave in his home, and Major Healy needs to do so as well.
Then, there's the result of whatever information the Colonel and NASA gained about her. Knowing that there were genies, NASA would divert funding to acquiring more, and using them to obtain better craft, and NASA dominance in the space race.
Not to mention the dramaturgical reason: by having a nominally friendly faced superior who needs to be kept in the dark, it creates many opportunities for great dramatic and humorous moments.
According to legend (granted some legends may vary) the owner of the lamp was the master of the genie bound to the lamp, and when it comes to genie lamps, ownership can be obtained by whatever means necessary, theft by a higher ranking officer included. However in the TV show Major Nelson kept Jeannie's existance a secret not because he was afraid Dr. Bellows would take her, but rather because he feared he would have been drummed out of the Air Force and the space program for having a magic genie.
As far as alien I don't think so she's EarthBound meaning she's from Earth. Well she would be a security risk and and illegal alien I think NASA would see the benefits of having a genie at their disposal. Even though they were paranoid they will be able to observe Major Nelson in Genies actions and find out that Major Nelson Has majority of control over her. Most likely what would happen is they would all 4 Major Nelson the choice whether to get rid of Genie and send her back where she came from or leave the space program and possibly face charges of a national security risk. Due to Major Nelson's excellent see in the astronaut program he would be given some leniency. Although slavery wouldn't be a charge because he said Genie free when he first got her it's by her choice that she stayed. Most likely she would be deported or going through the process of getting her green card. However knowing how evil and messed up humanity is Kama maybe let you know soon actually has a legitimate reason for keeping Jeanne a secret. However that's assuming that Genie allows the government to do that. Remember she has the ability to make everyone forget what happened. Remember major Healey when he when they found out he had all this expensive stuff. However even the genie has to follow rules. If you remember after the series there was a movie where Heidi gave Jeanne a choice... let Tony Nelson die or Genie has to leave Major Nelson and all of his memories have to be wiped away of her get Genies cannot marry humans. But I'm going off topic here the point is it all depends but most likely there's the worst case scenario or NASA would find some use for her. Human beings are like a coin. there's heads evil and Tails good Humanity. Too bad most humans I choose heads
Because it was the 1960's, Barbara Eden couldn't show her belly button, and it was unbecoming of an officer to live with (and own!) a beautiful, scantily clad, nubile, aryan woman that he wasn't married to. Nobody would have believed for a second that he wasn't doing what most of the male audience at the time dreamed of.
In fact, that was part of the gag, that he actually was an honest, upright, all-American hero who would never take advantage of the situation, despite the genie doing everything she could to get him to do so.
Especially because she really genuinely believed herself to be his slave, a very anti-American sentiment to be applied to a woman as described above. Do recall, this was at a time when the term "white slavery" was still in common use, and essentially meant white folks (generally women) kidnapped and sold into sex slavery- and this would appear very, very much like that.
Note that if he was merely dating her, this would likely be acceptable mostly (except the scantily clad bit). But two single folks of the opposite sex living together? Clearly living in sin. And thus unbecoming of an officer.