In The Desolation of Smaug, Beorn mentions to Gandalf that he remembers the time when Sauron ruled over Mirkwood.

[Beorn to Gandalf:] "I remember a time when a great evil ruled these lands. One powerful enough to raise the dead. If that enemy has returned to Middle-Earth, I would have you tell me."

Since Sauron was defeated 3000 years ago when that discussion takes place, that would mean that Beorn and his kin had lifespans measured in millenia, not unlike that of Elves.

Was it ever mentioned what the lifespan of a Beorning is and how long had Beorn himself lived?

It is unclear, even to Gandalf (at least as of The Hobbit, before the wizard and Beorn became friends) whether Beorn is fundamentally a man who can change into a bear, or a bear that can shapeshift into a man. As Gandalf puts it in "Queer Lodgings":

Some say that he is a bear
descended from the great and ancient bears of
the mountains that lived there before the giants
came. Others say that he is a man descended
from the first men who lived before Smaug or
the other dragons came into this part of the
world, and before the goblins came into the hills
out of the North. I cannot say, though I fancy the
last is the true tale. He is not the sort of person
to ask questions of.

At any rate he is under no enchantment but
his own. He lives in an oak-wood and has a
great wooden house; and as a man he keeps
cattle and horses which are nearly as
marvellous as himself. They work for him and
talk to him. He does not eat them; neither does
he hunt or eat wild animals. He keeps hives and
hives of great fierce bees, and lives most on
cream and honey. As a bear he ranges far and
wide. I once saw him sitting all alone on the top
of the Carrock at night watching the moon
sinking towards the Misty Mountains, and I
heard him growl in the tongue of bears: ‘The
day will come when they will perish and I shall
go back!’ That is why I believe he once came
from the mountains himself.

Although "came from the mountains" is ambiguous, to me this makes it sound as if Beorn himself may have been driven down out of the Misty Mountains when they became inhospitable—not merely that he is descended from men or bears who previously lived up among the heights. The Misty Mountains had been infested with orcs since at least the 2790s, when the War of the Dwarves and Orcs took place. Since Thorin and Company meet Beorn in T. A. 2961, Beorn seems to have been well over 150 at the time of The Hobbit. That is not beyond the limit of human lifespans in Middle Earth, but it does suggest that (beyond his ability to change shapes), Beorn is not a normal human.

On the other hand, we know from The Fellowship of the Ring, "Many Meetings":

Frodo learned that Grimbeorn the Old, son of Beorn, was now the lord of many sturdy men, and to their land between the Mountains and Mirkwood neither orc nor wolf dared to go.

From this it seems likely that Beorn was deceased by the year 3018, when Frodo arrived in Rivendell. Moreover, the fact that his son was given the cognomen "the Old" suggests again that Beorn's descendants also enjoyed longer lifespans than ordinary men.

The fact that Beorn seems probably to have died is also significant in another way. In general, the idea of intelligent, speaking animals (be they bears, wargs, ravens, or other) is actually somewhat difficult to reconcile with Tolkien's cosmogony, in which only those races given the Secret Fire by Eru possess free will and souls. Talking animals—both good and evil—crop up repeatedly in The Hobbit though, so there does need to be some place for them. It is sometimes suggested that, like the eagles (the emissaries of Manwë) all the other animals who talk—from Huan the Hound to the thrush who tells Bard of Smaug's weak spot—are actually Ainur in animal shapes. However, this seems not to be a viable explanation for Beorn, if did eventually die of old age.

No idea...

Movies

In the movies, Beorn mentions that he was from the mountains, and he is aware of the darkness in Mirkwood. Obviously at that time no one knew who the Necromancer was, so he couldn't have known it was Sauron.

[Thorin:] “You know of Azog? How?”

[Beorn:] “My people were the first to live in the mountains, before the Orcs came down from the north. The Defiler killed most of my family, but some he enslaved.”

[Bilbo sees the remnants of manacles on Beorn’s wrist.]

[Beorn:] “Not for work, you understand, but for sport. Caging skin-changers and torturing them seemed to amuse him.”

[Bilbo:] “There are others like you?”

[Beorn:] “Once, there were many.”

[Bilbo:] “And now?”

[Beorn:] “Now, there is only one.”

[The dwarves, Bilbo, and Gandalf look on in silence.]

[Beorn:] “You need to reach the mountain before the last days of autumn?"

[Gandalf:] “Before Durin’s Day falls, yes.”

[Beorn:] “You are running out of time.”

[Gandalf:] “Which is why we must go through Mirkwood.”

[Beorn:] “A darkness lies upon that forest. Fell things creep beneath those trees. There is an alliance between the Orcs of Moria and the Necromancer in Dol Guldur. I would not venture there except in great need.”

Books

Neither does he mention this in the books as well.

...but presumably as long-lived as regular, mortal Men

In Appendix F:

Most of the Men of the northern regions if the West-lands were descended from the Edain of the First Age, or from their close kin. Their languages were, therefore, related to the Adunaic, and some still preserved a likeness to the Common Speech. Of this kind were the people of the upper vales of the Anduin: the Beornings, [...].

So having no relations the the Numenoreans they did not have their extended lifespans, and therefore would have lived as long as Men normally would (presumably 80-100 years old).

tl;dr

Beornings are very long-lived for a man. You may expect them to live to 100-ish, but don't expect them to reach the 200-ish Dúnedain/Númenórean range.

Tolkien's Letter 144

To Naomi Mitchison
[Mrs Mitchison had been reading page-proofs of the first two volumes of The Lord of the Rings, and wrote to Tolkien with a number of questions about the book.]

(...) Beorn is dead; see vol. I p. 241. He appeared in The Hobbit. It was then the year Third Age 2940 (Shire-reckoning 1340). We are now in the years 3018-19 (1418-19). Though a skin-changer and no doubt a bit of a magician, Beorn was a Man. (...)

I'll elaborate a bit.

As to the movie - if you'd assume Beorn talked about remembering Sauron (i.e. essentially making him alive as early as Second Age), what he said would make him at least somewhat inhuman (compare to Elros etc.). OTOH, you may consider him to speak about Witch-king (or some other Nazgûl etc.) - Beorn wasn't a scholar, distinguishing between different evil supernatural beings wasn't his forte.

Alternatively, you can consider this one of the non-canon stuff that was created purely for the movie narrative, and has little in common with the real Tolkien's legendarium.

As to the books - currently, people can routinely live to 100-110 years if they take care of themselves, if they have good genes, and if they possess a bit of luck. Extreme examples reach about 120 years... and Third Era is long gone chuckle. Since much of Tolkien's works are actually a praise of simple way of living, a man living in harmony with nature would have been expected to also have a lifespan that's longer of what would you consider 'typical' for a man. Still, as @MatCauthon already noted - he's a man, a bit of a shaman-y shapeshifting "magician", with just an extremely long lifespan (ca. 150 years). His children would probably carry those longevity in them, although not in such extreme way (see e.g. Elros and his children).