What kind of form is “don’t anyone”?

Solution 1:

I've always found it more useful to think of categories like tense, number, person, &c not as attributes of particular word-forms but as attributes of the clause, which elicit particular word-forms in various contexts.

As John Lawler says, “it's still second person verb agreement because it's an imperative.” The subject of an imperative is implicitly ‘second-person’. In my terms, this amounts to saying an imperative elicits a subject which may be parsed as ‘second-person’.

But in English, this is scarcely a limitation at all. Only ‘personal pronouns’ are explicitly marked for person; all other pronouns, and all nouns, are unmarked and may be parsed as whatever person is properly demanded. (For that matter, imperatives often take a ‘null subject’, which isn’t marked for anything!) We may think of those other pronouns as ‘third person’, because that’s how we mostly use them, at least in writing. But if you require a pronoun of a particular class—a relative pronoun or an ‘indefinite’ pronoun, for instance—in a first- or second-person utterance, you don’t have to cast about for a second-person relative or indefinite pronoun; you just use the relative pronoun you’d use anywhere else, and the context defines it as second person. Note that this is not materially different from the way we use verbs, which are also mostly unmarked for person.

Of course if you have some reason to make the second person explicit, there’s a way of doing that, too:

One of you turn the heat down, some of you go shut down the warp drive, and don’t any of you turn your suit heaters on.

Your example does have another interesting agreement problem:

. . . don’t anyone turn his suit heater on.

You (or somebody) put that possessive in the ‘third person singular’, reflecting (at a guess) the pressure of a pronoun you almost always use in that person and number. I’d put it in ‘second person singular’, to reflect the context: Don’t anyone turn your suit heater on.

This is not a construction likely to arise in strictly formal writing, so it probably doesn’t matter; but maybe we should agree in advance that the most convenient way of handling it is the way we handle that other pressing pronoun-agreement issue:

Don’t anybody turn their suit heater on!

Solution 2:

Nice question :-).
"Do" is so useful and so used that most speakers do not notice the complexity of many of the constructs it ends up in. Sometimes they can be "unwound" by rearrangement, as here. Sometimes not.

Here

• "Please don't anyone turn his suit heater on"

is more easily understood if seen as meaning something like

• "Please do not turn your suit heater on, anyone."

ie It sounds like "don't anyone" + "do xxx"
but is in fact "don't do xxx" + "anyone"
= "do not do" + ...