Tony Stark, the original comics character, was originally based on Howard Hughes. As such his being a germaphobe backed up from this deleted scene for Iron Man where Pepper follows him around giving him hand sanitiser makes sense.
Interviewer: So, Iron Man, where does he come from?
Stan Lee: At the time I (laughs) at the time we did Iron Man I was really feeling a little cocky, I'm a little ashamed of myself, but it was in a time in the war and young people throughout the country hated war, they hated the military, industrial complex, hated everything and rightfully so.
So, I said I'm going to come up with a character who represents everything everybody hates and I'm going to shove it down their throats. You know I was younger then and what do you know when your younger? So I decided to come up with a guy who actually manufactures armaments, he's a multi-millionaire, I fashioned him a little bit after Howard Hughes. And I fashioned him a little bit after me also, he was irresistible to women, you know, the usual. No but I wanted him to be very wealthy and of course like every Marvel hero he had to have an Achilles' heel.
So I figured we'd give him a weak heart, that sounds bright, nobody had a weak heart at the moment. And I had to find a reason for the weak heart so we sent him over seas to where they're having a war and he gets something, a shell stuck in his heart and he needs iron armour to keep his heart beating, you probably already know the story.
The film version was then also further based on Elon Musk as Jon Favreau states in this piece he wrote for Time.
Elon Musk makes no sense — and that's the reason I know him. When I was trying to bring the character of genius billionaire Tony Stark to the big screen in Iron Man, I had no idea how to make him seem real. Robert Downey Jr. said, "We need to sit down with Elon Musk." He was right.
Time, Elon Musk By Jon Favreau
There are lots of theories that this has come about after the events at the start of Iron Man and him having his eyes opened to what his weapons are used for. However, I don't really think that's true. We only really see this pet peeve of his a few times throughout the series, if I'm remembering correctly, and they can all be explained away without having to delve into his psychological state.
In Iron Man 3 he refuses to take the business cards from Aldrich Killian.
He's still very much at the height of his playboy billionaire lifestyle where it's all about partying. A strange man, who would be odd to most, has just gotten into a lift with him accosting them and shoving their ideas on him. Tony is being a jerk and has already dismissed him. He doesn't take the card, not because of a peeve but because he doesn't want to give credit to this man.
Of course this is also likely the hundredth time this has happened today. As an extremely rich man who is also one of the most famous inventors everyone would want to talk his ear off with their new idea that will revolutionise the world. You soon learn that most of the time these inventions just aren't worth the time of day and so it is better to ignore them.
In Iron Man 2 he refuses to take the summons from the Marshal.
She's giving him a summons to a court hearing about him handing over his suit. He thinks this is a ridiculous thing to even have to think about never mind attend. Using the "doesn't like to be handed things" excuse is a way to avoid taking the summons and so ignore what is happening. Just look at his entire attitude in the court hearing, he's playing it all off as ridiculous because he knows it is; a sham to get him to give up his tech.
In Iron Man 2 with getting the man to put the strawberries on the seat for him.
This is the only one that isn't directly clear. However, I think this ties into the main reason for all of this, he's a billionaire and a celebrity. It's just a way of keeping control and staying above others which he loves to do throughout the MCU. It also could be a small eccentric quirk but with all the other things we see him do with the girls and the partying, that goes against this somewhat.
In The Avengers he refuses to take the file from Coulson.
Again this is a bit of a control thing, he doesn't want to be interrupted at the moment with Pepper. They are celebrating, he's enjoying himself, he just doesn't want to ruin the moment with work. When Coulson rings he uses the "Life model decoy" thing as a way to ignore it, albeit unsuccessfully. Pepper takes it off of Coulson because she thinks Tony is being daft and that he should look over the file as soon as possible.
There doesn't appear to be any greater meaning to this into his psychology and what he saw in Afghanistan. Sure that set him on the straight and narrow but it doesn't appear to have influenced his "handing him things peeve". Every case we see this happen can be explained away under the circumstances that it happens in. Or it's just a general don't hand me things because it comes with obligations attached like with Aldrich. This happens so much that it eventually bleeds into everyday life as well, easier to ignore being handed everything than be selective, that just gives people the chance to slip things in with what you do hand them.
I also want to point out the point made in Flater's comment. I tried to make this point myself on the Strawberry Man point but I didn't explain it well. His comment explains it better than I can though:
Also note Tony's inherently dominant personality, which is often verbal (interrupting, wisecracking, being willfully obtuse, taking the spotlight), but is in principle no different from how he approaches physical interaction: on his terms and no other way.
