Even if Stark sued the government and won, the most he could have received is reasonable compensation for his costs and/or damages.
Under US patent law , he can't stop the US government from using a patented product if it so chooses.
It is also possible Stark had not patented some of the suit components.
In order to receive a patent , you must make the details of your invention public.
The invention could then be copied, for example by supervillains who do not care about being sued for patent infringement.
Instead, Stark might effectively be keeping the suit as a trade secret .
In that case he would have even less protection against the government using the suit's technology.
(I Am Not A Lawyer, but a quick Google indicates the law is pretty clear on these points.)
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He may or may not have sued.
Lawsuits, especially between large corporations or the government, drag on for years and years through the various trials and appeals.
It is likely that if a lawsuit was in progress, by the time of the events of Iron Man 3 it was still in progress.
Just because it was not mentioned in IM3, doesn't mean it wasn't going on.
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You can't sue the US federal government unless they consent to being sued, under the doctrine of Sovereign Immunity.
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The suit wasn't stolen.
Watch the movie more closely, during the donut shop scene it was very subtlety revealed that Tony Stark allowed Rhodey to take the suit because it was his legacy.
Tony was under the impression that he was dying, and it was also revealed in the same scene that he was just giving stuff away because of that.
It was stated that there were security protocols that prevented unauthorized usage, but Rhodey was able to access and use the suit because Tony made him an authorized user for when he died because, like I stated before, he was under the impression that he would die very soon.
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What does being patented have to do with it?
Theft is theft.
Patents, and trade secret law would more come in to play when legally going after Hammer for stealing his design.
As far as legally going after the government for taking the armor, he wouldn't get very far.
The government would claim eminent domain.
Eminent domain can be for public safety concerns which the armor surely is.
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When James Rhodes took Tony Stark's armour in Iron Man 2, wasn't that stealing a patented product and if so why didn't Stark take legal action against the US government?  Pepper Potts was heard speaking to a lawyer about it when Stark visited her in her office but we know nothing ever came of that since the government still has the armour in Iron Man 3!
Why didn't Stark sue the government for stealing his armour in Iron Man 2
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<marvel><marvel-cinematic-universe><iron-man><iron-man-2>
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During the donut shop scene it was very subtly revealed that Tony Stark allowed Rhodey to take the suit because it was his legacy. Therefore, the suit wasn't stolen. Even if Stark was able to sue the government and won, the most he could have received is reasonable compensation for his costs and/or damages. The fact is Stark allowed Rhodey to take the suit so the suit technically wasn’t stolen. Moreover, the US Government cannot be sued unless they grant the person suing them consent to sue them. 
The fact is Stark allowed Rhodey to take the suit so the suit technically wasn’t stolen. The US Government cannot be sued unless they grant the person suing them consent to sue them. 