From a rules perspective: no
While this isn’t stated anywhere explicitly, we can infer it in the fifth edition from the rules for Monsters.
Of note, only adult or ancient metallic and gem dragons have the Change Shape ability, so most dragons capable of using it are already in the older age categories. This makes it a non-starter for the purposes of aging out of the less powerful younger categories.
The rules for the Change Shape ability - and for similar ones used by other creatures, and indeed for spells like polymorph etc - tell us what does and does not change for a dragon or other creature who uses it. Here’s the relevant second paragraph of the ability used in the stat blocks for ancient metallic dragons in the Monster Manual:
In a new form, the dragon retains its alignment, hit points, Hit Dice, ability to speak, proficiencies, Legendary Resistance, lair actions, and Intelligence, Wisdom, and Charisma scores, as well as this action. Its statistics and capabilities are otherwise replaced by those of the new form, except any class features or legendary actions of that form.
The changes are thus limited to those noted in the stat block of the new form, with the exceptions stated. Lifespan and a “rate of aging” are not features of stat blocks, and are not noted at all for most beasts. Most humanoid stat blocks leave open the question of which race they belong to, meaning that how they age is explicitly not part of the stat block, since different humanoid races can have very different lifespans and rates of aging.
From a lore perspective: also probably no
This is also not stated anywhere explicitly, but in the write-up for metallic dragons it mentions that some choose to spend considerable time in humanoid form to immerse themselves in other cultures. There’s no mention of this affecting how they age, which would be a significant limitation on this behaviour, so it seems unlikely to be the case. There is also at least one example of a long-time disguised dragon in a published adventure which makes no mention of any such problem with aging (I’m not being explicit about this one as it might be considered a spoiler).
The closest to an official lore answer appears in Fizban’s Treasury of Dragons, where at the end of the section on shape changing it mentions that some dragons believe if they spend too long in another form they may be trapped in it:
This is probably just superstition, but many dragons claim to know of others of their kind condemned to spend the rest of a very long life trapped in Humanoid or Beast form.
This suggests they wouldn’t age and die as a human or beast would, but live out their normal dragon lifespan of centuries.