Do the Dragons of Tolkien's works die a natural death, or are they like the elves and can only be killed in battle?
In 1960 Tolkien began rewriting The Hobbit in an attempt to harmonize it with the more developed story and history presented in Lord of the Rings, but didn't get further than part-way through the third chapter before abandoning it. This rewriting remained unpublished until it appeared in John Rateliff's History of the Hobbit, although it's existence was known of and it was referred to in passing in Humphrey Carpenter's Biography.
Among the changed passages was the reference to draconic lifespan in chapter 1, and here (in the final version of the text) we read:
To a mortal Hobbit a thousand years (which was an amendment from five thousand in an earlier version) might certainly seem as though it was "practically forever", and although this is a conjectural interpretation of the original wording, it does seem a valid one.
This is the only statement I am aware of that sets an actual definite and measured lifespan for dragons. It's dating (i.e. to 1960) is significant as it therefore can't be rejected as earlier and superseded material, but must instead be representative of Tolkien's thinkings in the time following the publication of LotR.
Thorin's speech at the beginning of The Hobbit suggests that it might be possible for dragons to die a natural death:
On the other hand, that could just mean that it's possible so far as Thorin knows. Tolkien never actually writes about a dragon who dies of old age. Certainly Smaug's boasting suggests that dragons get stronger as they age: