As Aspects, the dragonflights only tend to get involved when there is dire risk to Azeroth as a whole.
The First War only involved the Orcs and Humans and was limited in theatre to the kingdom of Stormwind. Other than Medivh going loco and screwing around with space-time - which one of the few remaining Blue dragons, Arcanagos, did try to intervene on - they had no reason to get involved.
The Second War raged all over what is now called the Eastern Kingdoms, and the Black dragonflight was involved behind the scenes, so the Reds would have been involved then. Unfortunately, Deathwing had already arranged to put the Dragon Soul in orcish hands, have Alextrasza captured, and her young enslaved. What remained of the Reds was busy trying to get her free. By the time they did, the orcs were on the run.
The other major event (by dragon reckoning) that happened at the end of the Second War was the destruction of the Dragon Soul. With all the Aspects now free to act, they all spent their time trying to hunt down Deathwing. We're not sure how long that took - at one point he was hiding on the orcish homeworld Draenor, and when forced to retreat back to Azeroth, he eventually ended up hiding in Deepholm, the pocket dimension for earth elements, where he still had some few allies.
That hunt is the only real explanation to the dragon's lack of involvement in the Third War. Most of the dragonflights were rebuilding their numbers, true, but there's nary a dragon to be found. The undead plague enabled a new invasion from the demonic Burning Legion, most of the planet was a battleground of one kind of another, and there was every reason for their intervention to be called for. It's possible with Archimonde wandering free that they could have lost if they showed themselves, but there's still no attempt to help even on the fringes of the war, or at the last stand at the World Tree. The only mention I can find is that Alextrasza and Ysera were asked in the aftermath to restore the World Tree, and refused, saying it would heal in time, leading the night elves to mangle an attempt to grow one on their own.
It's possible one of them was supporting Medivh as he was running around the Third War being mysterious, vague, not-insane, and not-dead, trying to get all the races to come together, but that's just one possibility among many for that particular enigma.
To consider the outside context rather than the lore, it's also entirely possible that the develeopers simply didn't think to put (or couldn't fit) dragons in Warcraft 3, which is where they're missed the most. Day of the Dragon, the novel that really exapanded the dragonflight lore, was released in 2001, and Warcraft 3 was released in 2002. The game was under development a lot longer than the book was, I wager, and it's not the first time different parts of a franchise haven't been speaking with one another. That just doesn't make for a very good lore-wise answer.
