Tolkien makes it clear in his letters that despite his remarkable qualities, Beorn was nonetheless a Mortal Man. Assuming that his lifespan was typical of ordinary Men, it's unlikely that he survived much beyond III 3000. (see The Letters of J.R.R. Tolkien, No. 144, dated 1954).

In the same comment referenced above, Tolkien confirms that despite his abilities, Beorn definitely belonged to the race of Men: 'Though a skin-changer and no doubt a bit of a magician, Beorn was a Man'.

The etymology of Beorn's name is interesting. The Old English word originally meant 'bear', but evolved over the centuries to the point where it came to mean 'warrior'. So, Beorn's ability to transform from a bear to a man echoes the linguistic development of his name.

The original meaning of Beorn relates to the bear's love of honey, and contains béo, meaning 'bee'. This also appears in the name Beowulf: like Beorn, this means 'bear', but is literally 'bee-wolf'. This bee connection explains the references in The Hobbit to Beorn's bee-pastures and huge bees: 'The drones were bigger than your thumb, a good deal, and the bands of yellow on their deep black bodies shone like fiery gold' (The Hobbit 7, Queer Lodgings).