It's important to be clear about what you mean by "benefit," but if you're truly obtaining no legally cognizable benefit, then a lie about citizenship status is protected by the First Amendment. The government may not use it as the basis to take any action against you.

Under U.S. v. Alvarez, 567 U.S. 709 (2012), the fact that speech is false is not enough to allow the government to legislate against it. Instead:

The First Amendment requires that the Government’s chosen restriction on the speech at issue be “actually necessary” to achieve its interest. ... There must be a direct causal link between the restriction imposed and the injury to be prevented.

If the government were to try to punish the lie you're talking about, the courts would ask (1) whether there was a compelling governmental interest in restricting that speech, and (2) whether the government had found the narrowest way possible to do so.

In this case, the government would fail at step 1, and if it didn't, it would definitely fail at step 2.

Step 1: Lies about citizenship may have all sorts of consequences that the government has an interest in preventing (in the areas of employment, tax, and immigration, for instance), but it doesn't have any interest in prohibiting lies by which someone obtains no benefits whatsoever, and which therefore implicate none of those interests.

Step 2: If a court said that enforcing immigration or employment laws was a good reason to restrict lies about immigration, it would still find that the government had not found the narrowest possible way to enforce the speech restriction. In cases like this, courts routinely conclude that a law that punishes harmless speech in order to prevent harmful speech is not narrow enough, as the government can simply write the law to punish only the harmful speech.

So the lie would not affect a person's status. If they were not removable already, the lie would not make them removable. If they were removable, the lie would not make them any more or less removable.

Of course, all this assumes that you are not using too narrow a definition of "benefit," which the government would probably construe quite broadly. If you were not attempting to get a government benefit, but were trying to get a job or vote, for instance, either of those would be considered a "benefit" that the government of the type that the government would likely have an interest in regulating.

It's important to be clear about what you mean by "benefit," but if you're truly obtaining no legally cognizable benefit, then a lie about citizenship status is protected by the First Amendment. The government may not use it as the basis to take any action against you.

Under U.S. v. Alvarez, 567 U.S. 709 (2012), the fact that speech is false is not enough to allow the government to legislate against it. Instead:

The First Amendment requires that the Government’s chosen restriction on the speech at issue be “actually necessary” to achieve its interest. ... There must be a direct causal link between the restriction imposed and the injury to be prevented.

If the government were to try to punish the lie you're talking about, the courts would ask (1) whether there was a compelling governmental interest in restricting that speech, and (2) whether the government had found the narrowest way possible to do so.

In this case, the government would fail at step 1, and if it didn't, it would definitely fail at step 2.

Step 1: Lies about citizenship may have all sorts of consequences that the government has an interest in preventing (in the areas of employment, tax, and immigration, for instance), but it doesn't have any interest in prohibiting lies by which someone obtains no benefits whatsoever, and which therefore implicate none of those interests.

Step 2: If a court said that enforcing immigration or employment laws was a good reason to restrict lies about immigration, it would still find that the government had not found the narrowest possible way to enforce the speech restriction. In cases like this, courts routinely conclude that a law that punishes harmless speech in order to prevent harmful speech is not narrow enough, as the government can simply write the law to punish only the harmful speech.

So the lie would not affect a person's status. If they were not removable already, the lie would not make them removable. If they were removable, the lie would not make them any more or less removable.

Of course, all this assumes that you are not using too narrow a definition of "benefit," which the government would probably construe quite broadly. If you were not attempting to get a government benefit, but were trying to get a job or vote, for instance, either of those would be considered a "benefit" that the government of the type that the government would likely have an interest in regulating.