Since Tony doesn't ever go and get actual treatment in the film, there isn't a clear-cut yes/no answer to this. You can read the DSM's criteria for PTSD here, and see that Tony definitely shows many of the symptoms of PTSD during the film. This article (despite frequent formatting issues) does an excellent job of discussing Tony's arc through the 4 films, and how they relate to PTSD. The article doesn't try to straight-up diagnose Tony, but does make several good points:
With fictional patients, labels can normalize the experience of disorders and destigmatize mental illness. But the brilliance of the third Iron Man installment rests in the complexity of Tony’s psychology. Iron Man doesn’t work in absolutes either. The movie seems to intentionally depict mental illness with a sophisticated level of ambiguity and dimension, emphasizing the realistic point that the course of mental illness has no clear-cut causes, cures, beginnings, or ends.
This praise for a complex, realistic portrayal of Tony's issues matches the praise I've seen elsewhere for Iron Man 3's inclusion of mental health issues. For example:
It was amazing to see the number one movie in the world, a huge summer blockbuster, handle a mental health condition so calmly and rationally. Everyone in Tony’s life knows he’s sick, and they try to help him. He does what he can to mitigate his illness (though he never gets treatment, as far as I could tell), and to get better. No one questions his ability to be a superhero or a genius or even a functioning person. The illness itself was shown as life-disrupting and scary for him, but not so over the top that it was cartoony.
Whether he does or doesn't have PTSD specifically, he has some severe mental health issues, and one of the film's arcs is his coming to grips with it. The way he talks though, I think makes it clear that his issues stem from what happened in New York during The Avengers, and not the future threat we the audience know will come from Thanos. Tony telling Pepper that he can't sleep because of an imminent threat sounded to me like deflection. He knows that thinking about and talking about New York is a trigger for his anxiety attacks, by saying he's building the suits for some vague future threat he sidesteps discussing New York and the pain that comes with that.