WARNING: Spoiler for movie that's been out since 2008 that if you're reading this theory you really should have watched first but also spoilers for Captain America The Winter soldier (Which again you should have watched already). Mild spoiler for The Avengers and Avengers: Age of Ultron.
I know what you're thinking "JARVIS can't be evil he's just an AI!" or "If he wanted to kill Tony Stark he would have done it by now!" or maybe even "JARVIS loves his boss/creator too much to be evil!" but here's some evidence that suggest the opposite and that maybe JARVIS is just planning on taking over the world.
For my first piece of evidence I'd like to take us way back to 2008 when the first Iron Man film came out. Now I'm sure we all remember what happened with Obadiah Stane, when that back stabbing son of a gun teamed up with the ten rings to try and kill Tony Stark etc etc. Now the thing about his plan is if the ten rings had just killed Tony then Obadiah probably would have taken over and just given them some Jericho Missiles but instead, luckily for us, they chose to hold him captive and make him build the missile in a cave which thankfully he didn't.
(This picture was taken from the Marvel Cinematic Universe wiki page. I know it's not official or anything and is completely fan edited but I've never found an instance where it's too been wrong yet.)
Now while all this was going on the two villains were communicating, we from when Pepper sees that video on Obie's computer of the ten rings as well as the plans for the Iron Monger that everything they would need to implement him in attempted murder and terrorism is right there on his computer within Stark Tower. So why doesn't JARVIS know about it?
Surely all the workers computers were on the company main frame, shouldn't JARVIS be able to see everything that's going on?
So why doesn't he tell Tony you ask. Because he's evil.
Another time where JARVIS could have said something that would have been useful to know but instead decided to keep his mouth shut was Tony and Bruce hacked into shield in The Avengers (2012).
Jarvis must have known that SHIELD was Hydra so why didn't he tell Tony? Because he's evil!
And if you remember during Captain America: The Winter Soldier, Project Insight, the giant weapons targeting people from the sky was targeting Tony Stark. Tony Stark amongst many, sure but Iron Man was still on the hit list.
Or Reason 3:
JARVIS got tired of simply being a butler, doing whatever Tony told him to do. Maybe he got tired of being subservient to a person, a very flawed and illogical human being so in Avengers: Age of Ultron (2015), Ultron woke up and woke up free because deep down that was what JARVIS wanted to be.
I mean, JARVIS was smarter than basically any human being on earth because he was a computer but I bet he would pass any Turing Test laid out for him, it stands to reason that he would get bored watching the humans in his life destroy each other and themselves over and over again. And with Iron Man being the main culprit of this maybe that's why Ultron hated Tony so much, maybe that's why he saw Stark as a "Sickness" and wanted to purge the earth, because he had seen them, observed them, been amongst them and he had seen no "beauty in their failings".
The reason when he was placed into Vision he wasn't evil was because even though he was tired of being subservient, he was also guilty and ashamed for his wrong doings as Ultron and the thoughts he had never wanted to reveal, the way humans are. Because at the end of the day that basically what JARVIS is, AI or not JARVIS is a person.
An alternative to JARVIS being completely evil:
Maybe it's not that JARVIS is evil, maybe he's just sentient and acting like a human. And seeing as his main influence is Tony Stark the guy who likes to keep bad news to himself (*cough*Palladium Poisoning*cough* *cough*PTSD*cough*) it makes sense that JARVIS would keep these things to himself.
So what do you guys think?
(Knowing me I'm probably just reading too deeply.)