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 This rewriting remained unpublished until it appeared in John Rateliff's History of the Hobbit, although it's existence was known of and it was referred to in passing in Humphrey Carpenter's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:
...they guard their plunder as long as they live, a thousand years maybe, unless they are killed...
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 It's dating (i.e. to 1960) is significant as it therefore can't be rejected as earlier and superseded material, but must instead be representative of Tolkien's thinkings in the time following the publication of LotR.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:
"Dragons steal gold and jewels, you know, from men and elves and dwarves, wherever they can find them; and they guard their plunder for as long as they live (which is practically for ever, unless they are killed)..."
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:
I laid low the warriors of old and their like is not in the world today. Then I was but young and tender. Now I am old and strong, strong, strong!
I may add that the Hobbit is a little early in Tolkien writing when he had not yet established a firm lore. If we take Glaurung as the prototypical dragon,it becomes more difficult. It's possible that he is a Maiar (although this is a contested theory) incarnated thus unable to really die.
'And Glaurung spoke by the evil spirit that was in him'
So, we should probably know more about the nature of dragons before being able to answer this question properly: are they Maiar? I tend to think they are; I can't picture such an intelligent race not being an Ainur or a Children of Eru. It's not within Morgoth to create life only to corrupt it, and again I can't picture anything lesser than a Maiar being corrupted into a dragon.