SETTING: You live on a planet inhabited by intelligent dragons with the physical characteristics of a fire-breathing Smaug-type dragon. They are not above only partially burning their victims before consuming them. Like the French, they like their meat bloody!
Adults dragons live in small groups, in isolated pockets. They lay several eggs in each clutch, hatching dragonets of the size of horses. Only one or two dragonets making it to full adulthood from each batch. Once matured, the young dragons leave the nest and join up with other zit-faced dragons forming larger social groups until their youthful exuberance amongst themselves has whittled down the young group size to their adult social group setting.
They have above human intelligence, excellent hearing, and incredibly tough scales, plate armor, muscles to get through if you intended on trying to kill them. They release pent up gases and heat as a steady stream of fire.
Mercifully, they normally stay out of the way of humans, and allow normal life to continue. So they have never really been a threat before, as long as you kept out of their way and didn’t mind a few cattle going missing every now and then.
SCENARIO: Your planet experiences the zombie apocalypse. Mercifully, you find out that ALL the ancient fables on how to kill a zombie are correct.
The virus affects the infected intelligence regions, making them 'dumber' but not completely stupid. The smarter you were before you were infected, the 'smarter' you are as a zombie. This allows for some really stupid Zombies and some not so stupid Zombies.
The virus also affects the infected individuals speed and co-ordination so that you get the slow shuffling type of zombie (the thought of a smart, fast moving zombie is just too scary to contemplate, so I'm sticking to slow and shuffling).
The Zombies lose any interest in things that interested them before and being attracted by sound are prone to forming vast zombie hordes (a zombie shuffling along is not the quietest thing in the world).
The virus is transmitted by infected blood mixing with uninfected blood. So just getting blood on you is not a death sentence. This also means that eating uncooked flesh or partially uncooked flesh transmits the virus and it has spread to the rest of the animal kingdom.
Eventually the top apex predators are infected. You find out the cure is in the dragon's [insert ‘body part that is hard to get to’ here]. Unfortunately, the dragons have now become infected. And any that aren’t infected, aren’t willing to give up their lives so that you can live. Besides, any dragons not infected are smart enough to flee to some extremely isolated spot and QUIETLY wait the apocalypse out. Time for a good snooze!
ASSUMPTION: There are human survivors, somehow.
QUESTION: How would you go about killing a Dragon Zombie?
With the cure in the [insert 'body part that is hard to get to' here], you can't just ignore the dragons and hope they don't come after you.
Taking into account it's MASSIVE SIZE, scales, thick skull etc. It can no longer breathe a steady stream of fire at you, but has to release the buildup of heat somehow and seems to hiccup a fireball every now and then.
Starting from an above human intelligence, an average dragon zombie is now about as smart as the average human. With a loss of coordination, it is as likely to crush you by accident as to try and get to your brains.
It has lost the ability to fly (with intentional takeoff, wing-flapping, and precision landing), but can climb with some effort. If pushed off the edge of a cliff, it could have sufficient time and brainpower to open their wings to manage a sort of glide and not crash to their deaths!
EDIT: the Zombies are of the slow shuffling 'Walking Dead' kind and not the energiser bunny 'World War Z' type!
FURTHER EDITs to improve Dragon population numbers, dragon description, zombie description, dragon zombie description.