The demonic virus that changes humans into vampires replaces the telomere structure with something else. It's why they don't age and part of why they can heal so fast. Some theorized that vampire telomeres were infinite, but this would mean (theoretically) that vampires would just keep getting bigger. They don't, though they do change. This is because the virus takes over their body and modifies their gene-structure to more closely resemble a creature of the Void. Most vamps die long before any significant changes occur, but very old vampires (300 years or so) start to look less like humans. At around 1000, they can't even pass for humans anymore.
Dragons don't have DNA unless they take human form, at which point their dragon body creates a mimic of human DNA. Their human forms age with their dragon form, as the energy that keeps them alive (and allows them to change) gradually wanes. Dragons in their natural form have a completely different biological makeup, something beyond the realm of human science and understanding.
That said, I'm not one of those writers who has to have a scientific explanation for everything in my books. I know all of what I said probably doesn't make any sense from a practical standpoint, but it's how it works in my books because they are fantasy and paranormal.