I'm going to go for this one before editing my answer on the previous one, because I was giving this some serious thought today while driving.
Feaurie Vladskovitz's answer hits a lot of the basic ideas that I'll be working off of: the dragon-knights are air superiority in a setting where air war is just getting off the ground, so they're going to represent a major threat that any other human kingdom in the setting will have to deal with if they plan to expand (or even just not be conquered). Before I look at how the other kingdoms (both enemy AND ally) might deal with this, I want to digress a moment into how the people of those neighboring kingdoms might feel about dragons as a weapon of war. This informs what their leaders might decide to do about dealing with the threat of dragons being used against them.
How do people feel about the Dragon Kingdom?
Let's look at some of the components going into this.
1. Fire is a really horrible way to die. And your dragons are fire-breathers, who can't be touched by conventional troops, and they can fly, which means they can appear anywhere above the battlefield to rain down hot, fiery death whenever they feel like it.
While fire has been used as a weapon of war from time immemorial, its primary uses during the time period we're looking at (a medieval-ish setting) were during siege warfare, in area denial campaigns, and in demonstrations of dominance over conquered territory. This isn't to say burning arrows and the like didn't make an appearance on the battlefield too, but use of fire was much less when there was a risk of burning your own guys to death. (Check out Wikipedia's page on early thermal weapons weapons for more on the use of fire in warfare.)
So until the advent of dragons in warfare, your average soldier in the military of one of these various kingdoms had a relatively small risk of death by horrific burning, unless he was involved in a siege. And, the risks he took there were known risks that could be mitigated with the proper precautions (the page above has more on that). That all changed when the Dragon Kingdom brought flying flamethrowers into play, and now entire companies are being torched left and right by a single guy on the back of a flying lizard.
Can you imagine how horrifying that would be? Even leaving aside the fact dragons themselves are terrifying to look upon, seeing one over the battlefield means you now have an astronomically higher chance of being burned to death, along with all your buddies, and there is nothing you can do about it. This is not to mention the potential use of dragons against farms, towns, or cities. Firebombing was rightly seen as an awful means of war during World War 2, and the dragon-knights would have the potential to implement it in your setting. Do they?
Even if they don't, there's a lot of reason to be afraid of the threat of dragons just on the grounds of the use of fire.
2. Flying is cheating. While it's said "everything is fair in love and war," which is largely true for most of human history, war has been tamed by the conventions of warfare. Some kinds of behavior are considered off-limits for a civilized army, and soldiers were expected to more-or-less not engage in these behaviors in order to limit the horrors of war. You may want to consider whether your setting has a moral entity--like an established church, or widely followed religion, or something else entirely--that deals with ideas like just warfare and the proper usages of war.
Regardless of how you choose to establish the law of war in your setting, the introduction of dragons to warfare will seem like cheating at a gut level, because nothing and no one has a really good chance of standing toe-to-toe with a dragon. Consequently, every kingdom but their own might regard the dragon knights as cowards, because how much courage does it take to ride in on an indestructible war-machine and torch a bunch of guys who can't even reach you?
Anyone--a brilliant mage, an incredibly lucky archer, a gutsy band of conventionals with a really clever idea--who could take them down would also be seen as a Big Damn Hero by their own people, because in the early stages of the use of dragons in warfare, the damn things seem pretty much invincible!
3. Those aren't even humans! Nor are they domesticated animals, like horses or elephants or anything else mankind have tamed and ridden into war. They're fully sapient members of the Dragon Kingdom's community, and let's face it, that's a little bit weird. Or a lot weird. Especially if the sovereign dragons are largely regarded by most humans as a special sort of talking animal, so the inclusion of the tame dragons into human society looks kind of like letting Mr. Ed vote in local elections.
This will undoubtedly lead to nasty rumors about the dragon knights and their kingdom in the surrounding communities, too. Just how nasty those rumors are, and what their content are, well--I'll leave that up to you. One rumor that absolutely will show up, though, is that the Dragon Kingdom is not being ruled by its human ruler(s). Given the power difference between humans and dragons, and the fact "tame" dragons are demonstrably not tame the same way a domesticated animal is, the simplest and most obvious conclusion as to why such a kingdom works is that the dragons are running it, not the humans.
Allied kingdoms might know better, in that they'd have more insight into the workings of the Dragon Kingdom--but they'd never be able to shake the niggling idea that the dragons are the ones in charge. Enemy kingdoms would incorporate this idea into their propaganda. If they're especially clever, they might even try to portray the average human citizen of the Dragon Kingdom as laboring under the yoke of the oppressive dragon knights (or might even believe that).
PHEW. Sorry. That was a lot, and I can think of several more things to cover. I will leave those for now unless you want all the gory detail!
So what should be done about the Dragon Kingdom?
Or, how do we get rid of those damn dragons and bring war back to an equal footing?
While these are mostly written with the enemies of the kingdom in mind, its allies will also be keeping these ideas in mind...and maybe working on them on the side, too. Even if the Dragon Kingdom has treated its allies with flawless courtesy in the past, it has an incredible weapon now, and there's no telling when it might decide to take over the rest of the empire for funsies.
1. Air support of our own. Feaurie touched on this one already, so I won't go into too much detail, except to add another possible option: Other kingdoms might try stealing sovereign dragon eggs or young to try and replicate what the Dragon Kingdom has. Or, they might just go to the tame dragons (or even the dragon-knight pairs) directly and try to get them to defect. The Dragon Kingdom might want to keep an eye on dragon immigration, if they even allow it, because that could be the seed of a new dragon air force going to the hands of their enemies.
2. Get rid of the damn things. Dragons might be difficult to take down on the field of battle, but that's not the only way to get rid of them. Poisoning their food supplies, introducing a plague that targets dragons, or simply trying to make the political situation in the Dragon Kingdom suddenly inhospitable are all potential options. Another option would be to target the Dragon Kingdom economically, and make it much harder for them to support dragons. (This is where it will be good to know what and how much dragons eat. Since they're warm-blooded, flying megafauna, I'm assuming "a lot and mostly meat". If the Dragon King doesn't itself have the required infrastructure to produce all that food, they're probably importing some of it, which is an economic weak point to be exploited.)
If dragons won't fly in battle without their knights, then target the knights as well. Send assassins after them, disrupt their command structure, and make it impossible for the dragon teams to get onto the field.
3. Develop better anti-air capabilities. If somebody figures out a way to take out a dragon knight with magic (or massed archery, or anything else), you can bet knowledge of that will propagate like wildfire through that kingdom's military. Those kingdoms that do work out anti-air tactics might even share them, since it benefits everyone if dragons are used less in combat.
4. Make friends. Either ally with the Dragon Kingdom, or against them. If all that's keeping them from going wildly expansionistic is the presence of too many potential enemies, it's a good idea for you to make friends with all those potential enemies so you can reinforce each other if you're attacked. And, if the sovereign dragons aren't on a really good standing with the Dragon Kingdom, it might make sense to offer some kind of mutual protection pact that wouldn't hurt their draconic pride. They don't need to suffer themselves to be ridden into battle like beasts of burden--instead, for a regular cattle-tithe and the guarantee they won't get shot at by Xyzland's military, they can focus on taking out the dragon knights whenever they show up to fight...