Why isn't "witness" the third-person singular form in the example sentence?

I think this usage is an imperative, of the suggesting type. This dictionary agrees:
Merriam-Webster witness

verb 4b : to take note of
our grammar—witness our verb system—is a marvel of flexibility, variety, and exactitude— Charlton Laird

AHD witness

2b b. To consider as an example. Often used in the imperative: Even a widespread species can go extinct. Witness the passenger pigeon.

If you substitute the word note for the word witness in the OP example, the meaning and the feeling of the sentence are about the same.

Authentic Italian cooking is very healthy —— note the low incidence of heart disease in Italy.

One might interpret the usage differently, as if it were a subjunctive form, followed by its subject. : -witness the low incidence of heart disease =
-[let] the low incidence of heart disease [be] witness

Or one might call it an idiom. But in any case, the usage is valid - witness its commonness.

To witness something is to observe it. The sentence could be paraphrased

Authentic Italian cooking is very healthy —— observe (as evidence) the low incidence of heart disease in Italy.

"Look at the low incidence .." is another, less formal paraphrase.

In many contexts "witness", "observe" and "look at" have slightly different meanings, but here they are equivalent.

The sentence Authentic Italian cooking is very healthy —— witness the low incidence of heart disease in Italy. is an example of parataxis: two short complete sentences that are combined without any conjunction. Without changing anything, you could actually rewrite this sentence like so:

[Authentic Italian cooking is very healthy.] [Witness the low incidence of heart disease in Italy.]

The first sentence Authentic Italian cooking is very healthy is a normal declarative sentence, but more importantly, the second sentence Witness the low incidence of heart disease in Italy is indeed an imperative sentence (as @Jack-oflaherty's answer says). As such, it doesn't have a subject, and like in all English imperatives, the verb occurs in its base form without any inflectional suffix.

This answers your questions:

1. What is the subject of witness in the sentence?

Authentic Italian cooking is the subject of the first sentence Authentic Italian cooking is very healthy. The sentence Witness the low incidence of heart disease in Italy doesn't have a subject, as English imperative sentences are subjectless.

2. Why isn't witness in the third-person singular form?

In English, imperative sentences are not marked for number, person, or tense, but feature the verb in its bare form (here: witness, not witnesses).

3. Can we take it as an imperative sentence?

Yes, absolutely – but only the second part witness the low incidence of heart disease in Italy, not the first part Authentic Italian cooking is very healthy. This is a declarative sentence.

By the way, your rephrasing changes the grammatical structure from parataxis to hypotaxis, or more conventionally, to a complex sentence with a subordinate clause:

[Authentic Italian cooking is very healthy [as witnessed by the low incidence of heart disease in Italy] ].

This type of subordinate clause (sometimes called passive-participle clauses) is also subjectless, and it uses the past participle form of the verb (as seen, for instance, in the sentence Authentic Italian cooking is very healthy as eaten by the natives in Italy, where the past tense form ate would not be possible).

Change the sentence from "Authentic Italian cooking is very healthy —— witness the low incidence of heart disease in Italy" to "Authentic Italian cooking is very healthy —— look at the the low incidence of heart disease in Italy"; that should make clear what happened.

"Witness" is both a noun for a person who observed something, and a verb describing the act of observing something. In this case, "witness" is the imperative form of "to witness".