At the end of Iron Man 3:
Tony Stark removes the shrapnel in his chest and the ARC reactor, and destroys all of his armours. However, he ends by saying: "I'm Iron Man"
Does this happen in any other canonical source? And if so, how does the story continue?
The Marvel film-verse (including Hulk, Captain America, Thor, & Iron Man movies) has its own canon, separate from the comics.
So the easy answer to your question is "Yes, this is canon for the films," as the film itself is a canonical source.
Outside of that, I believe Tony Stark has gone through many ups and downs in his life in the comics. Simply destroying the armors and the reactor won't stop him from creating new armor or new reactors in the future.
What I think you're missing is that Tony Stark is making a point - HE is Iron Man. The armor doesn't make him a hero, it's just a tool. He's making the same point that he and Rhodie made in Iron Man 2 - it's the person inside the armor that matters, not the armor.
It's based off the Extremis storyline in the Iron Man comic just changed a little so it fits with the movie and I think they purposely didn't explain everything so they can reveal the new extremis Tony Stark/Iron Man for Avengers 2.
Extremis Iron Man
After being critically injured during a battle with a nanotech-enhanced foe, Stark injected his nervous system with a modified techno-organic virus to save his own life. This fused Stark's armor to his body, allowing him to store the inner layers of the Iron Man armor in the hollows of his bones as well as control it through direct brain impulses. The Extremis enhancement has turned Stark into a cyborg, whereby the usage of his existing lockchip (a personal area networking implement implanted in his forearm) is directly integrated into his nervous system.
His new armor is no longer a bulky unit which houses its own AI "response server" and miscellaneous interfaces for neural control. Instead, it is more lightweight (constructed of a pliable crystalline material with a molecular structure that can collimate into super-hard planes upon the application of an electrical field) and less complex (as it interfaces directly to Stark's brain via the Extremis-modified cybernetic connections), and has much faster response time since it effectively functions like Stark's second skin.
The series has ended. The story does not continue in any way. Robert Downey Jr. and Gwyneth Paltrow have stated that the series ends with Iron Man 3. The arc thingy was never used in the comics. Also, he tossed the reactor away. Even if he does use the armor (if he builds one) with an arc reactor fitted inside the armor, it won't matter cause nobody is being stopped from just ripping the reactor away. He created-and used Iron Man. He still remains Iron Man without the armor. The answer to the Question: "Does the man make the suit or the suit makes the man?" is-the man makes the suit. The suit is just a tool, as Jeff said. Only Tony, inside the suit matters. The shrapnel was never removed in the comics because the heart itself was removed and replaced/repaired with artificial tissue. The whole series is a canon.
