What does a dragon do when it dies?

That depends on your meaning. If the dragon has transferred its consciousness into their eldunari, they will live on forever trapped in a little gem-like object unless it is destroyed. Awfully dull fate. They will suffer the loss of a functioning body. If you mean what the Rider does when their dragon dies, often times they will go mad or commit suicide out of grief. Vice versa with the Rider dying and their dragon surviving. Sometimes they still remain sane and continue to have a long life as in Brom's case. Brom was approximately 120-130 years old. Sometimes a Rider's dragon dies but the Eldunari lives on so they still have their dragon's company.