When did we learn that stars die?
Super novae were known a long time ago. But they were not understood as a the death throes of a star.
In spite of the apparent immutability of the heavens, Chinese astronomers were aware that new stars could appear. In 185 AD, they were the first to observe and write about a supernova, now known as the SN 185. The brightest stellar event in recorded history was the SN 1006 supernova, which was observed in 1006 and written about by the Egyptian astronomer Ali ibn Ridwan and several Chinese astronomers. The SN 1054 supernova, which gave birth to the Crab Nebula, was also observed by Chinese and Islamic astronomers.
But it wasn't until later we understood they had a life-cycle. The Greek philosopher Aristotle even proposed that the stars were made of a special element, not found on Earth, that never changes.
The Chinese might have been the first to suggest the idea as they took careful note of "guest stars" which suddenly appeared among the fixed stars.
It would seem that another person who suggested the idea was probably Tycho Brahe (14 December 1546 – 24 October 1601) as he coined the term nova meaning new star. And likely with this new mind set births brings deaths. He also is famous for realizing stars are very far away (due to parallax). In 1572 he witnessed a super nova and in 1573 he published a small book, "De nova stella" (The New Star) based on the super nova he saw. (Most super novae were assumed to be new stars, not dieing stars).
The event of understanding stars die probably just fell out of understanding what stars are. I'm not sure you can point to one event or person in history that could prove to know stars die prior to understanding stars themselves.
Conservation of energy dates back to ca. 1840. Once that was established, it was natural to suspect that a star had a finite lifetime, which could be calculated if its energy source was understood. The most popular theory in the 19th century was that stars converted gravitational energy into heat. Lord Kelvin also hypothesized that meteors crashed into the sun and resupplied it with energy. All of these mechanisms gave lifetimes for stars that were relatively short, and, e.g., too short to be consistent with the kinds of terrestrial timescales being proposed by Darwin and the geological gradualists. More info here: http://www.nobelprize.org/nobel_prizes/themes/physics/fusion/sun_1.html