I think we’ve established a consensus that most literary and mythical dragons are intelligent creatures capable of mourning for lost friends and family. Perhaps more interesting is how dragons would show their grief.
Since dragons are often depicted as powerful beings, you could have dragons taking part in human or similar funerals in some capacity. Perhaps to show diplomatic honor for allies, a dragon might stand as honor guard for the fallen. Dragons might fly over ceremonies in tribute, or even simply attend the funeral if space allowed.
A few books show a different approach to the passing of dragons: cannibalism! In some of Robin Hobb’s books, a dragon that dies will be eaten by other dragons. This is because dragons are so magical, anyone who consumes the dragon’s flesh will gain great power. The dragon’s flesh also contains their memories and insights of past generations. Dragons eat their dead in order to maintain draconic power and heritage.
There’s another book I recall from about 15 years ago, where cannibalism was the basis of draconic society. It was much like a Regency romance, except all the characters were dragons. Eating other dragons was essential to a dragon’s healthy growth, and there was a lot of intrigue by powerful and upwardly mobile dragons creating excuses to devour the young and weak of their own kind. When a dragon died, there was a strict hierarchy of who could eat what parts of the deceased. Major plot points revolved around a greedy relative who took more than his share.
That sounds really bizarre, now that I think about it! I’ve searched high and low and can’t find this book, so if you’ve ever heard of it, I’d love for you to share the information.
Actually, there’s been a lot of sharing on this topic. I want to thank all of you for your comments. So, how do you think dragons would mourn?
Wyrmflight 
I think you are speaking of Tooth and Claw, by Jo Walton.
Ah, yes. That looks right (I searched for it online). And it won a World Fantasy Award in 2004, too!
I’ve got my current dragon relating that he ate his clutch-siblings, but that was to demonstrate how alien dragons are in that particular setting.
For mourning a human companion, I could see the dragon forbearing another rider, perhaps forever, or perhaps marking itself in someway: a jeweled scale. I could also see it performing certain rituals, maybe giving the human a dragon burial that could involve stone tomb sealed by rock melted in dragon’s breath.
For mourning another dragon, evil ones might vent their anger on humans. Good ones might endow a shrine.
One concept I’ve toyed with over the years, in D&D and in stories, is that dragons don’t raise their young and young dragons are rather foolish, albeit powerful. Some dragons might then offer a whelp to a human to have the human train them in the ways of the world and mature them into adulthood.
In this scheme, the dragon might have only a single rider, whom he or she serves until the human’s death. The human’s death would then be both sad for the dragon, as the human might be something of a mother or father figure, but also a key rite of passage: now they are an adult.
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