The short answer is: accountability. After Obadiah Stane's betrayal during Iron Man 1, Tony doesn't trust anyone else with the Iron Man technology (except his best friend, James Rhodes), because he does not want to risk it being used for the sort of atrocities the Ten Rings used his weapons for in Iron Man 1. This is also the reason he shuts down Stark Industries' weapons division.
This idea of accountability, or of others getting their hands on the Iron Man tech, comes up time and time again throughout the Iron Man trilogy:
At the climax of Iron Man 1, Obadiah Stane builds and pilots the Iron Monger suit based on Tony's original prototype. This drives home to Tony just how dangerous the Iron Man tech could be in the wrong hands:
OBADIAH: Ironic, isn't it, Tony? You tried to rid the world of weapons; you gave it its best one yet. And now, I'm gonna kill you with it.
The Senate Committee scene in Iron Man 2 is the US government asking Tony to turn over the Iron Man suit, presumably so they can mass-produce them. Tony point-blank refuses, because he knows that's what they're planning to do and doesn't want to be held accountable for what the government might do with them. It also demonstrates that the Iron Man suit isn't as easy to mass-produce as you believe: look at how Iran and North Korea's attempts went. (And just to justify Tony's paranoia even more, Captain America: The Winter Soldier reveals that the head of the committee, Senator Stern, worked for HYDRA.)
- Then, later in Iron Man 2, the US military do get their hands on the Iron Man tech when James Rhodes brings them the Mk.II suit. Justin Hammer gets the contract to develop the mass-produced suits, and gets Ivan Vanko/Whiplash to assist him. This, of course, ends with Vanko hijacking the suits (including the one Rhodes is inside) and sending them on a violent rampage as part of his campaign to destroy Tony's legacy. This would only reinforce Tony's refusal to militarise the Iron Man program - what if those suits got hacked?
- Interestingly, in Iron Man 3, there actually sort of is an Iron Man army - Tony has built dozens of suits that can all be remotely operated by JARVIS. However, if I recall correctly, he didn't build them specifically to serve as an army in case of a serious threat, but as a coping mechanism after the events of The Avengers left him with PTSD. At the end of the film, having come to terms with his trauma and vowed to prioritise his relationship with Pepper Potts over his Iron Man career, Tony destroys the surviving suits.
It's been pointed out repeatedly in the comments that in Age of Ultron, Tony has another fleet of autonomous Iron Man suits, the Iron Legion. However, he uses them for crowd control to keep civilians safe while the Avengers do the actual battling. Presumably, he still doesn't trust others with the suits, and after Ultron hijacks the Iron Legion partway through the movie, I'm guessing he no longer trusts AIs with the suits either, hence why he hasn't rebuilt the Iron Legion since then. There's another autonomous suit in Spider-Man: Homecoming, but I believe Tony remote-controlled it directly rather than having an AI control it.