Magic is gone from the world, with the exception of a few members of a dying civilization. Plate and chainmail armors are commonly used by the soldiers in this world (a low fantasy medieval setting). He is entrenched in a castle, with many faithful followers surrounding him. He will easily be able to cook any knights alive that manage to break through the castle's defenses. He is relatively old, but the circumstances do not allow a protracted siege. It is not feasible to send in assassins, as the garrison and the mage's followers are relatively few in number and all know each other by name (around 120 in number). A direct assault would be folly, since many of his followers are also adept in the use of fire, and could simply turn the surrounding fields to ash. The mage's main weakness is water. What would be the easiest way to defeat him?
Nuke the sh*t out of the castle with flying rocks, you'll take care of him this way. If the mage's only power rely on fire, he won't be able to flee and wouldn't be able to melt 30-something rocks flying straight through the castle.
Find a water mage
If that's possible, find a mage that can counter his abilities.
Wait for rain?
If water is a true weakness (that negates his powers or something of this magnitude), rain would be enough to make him powerless and then you could charge the castle with everything you got.
If fire's his only power, he's as vulnerable to a siege as any other human, perhaps more so if he's the only inhabitant. So, as Samuel suggested in the comments, stay out of range, and encircle his castle in stone fortifications with round-the-clock guards, and just wait him out.
If you need him defeated more quickly than that, or he has the means (though magic or any other way) to stay self-sufficient within the walls of his keep, it gets harder. You'll need a material that is tough and fireproof but light enough to replace wood or metal shields. Dragonhide/dragonscale is a fairly common hand-wave, though typically too rare to equip more than a small band of heroes. In more real-world scenarios, tightly-woven wool stretched over wood or metal will insulate against heat and won't sustain fire, making for as good a shield as you're likely to get. Doesn't work as well for armor; a shield will more easily dissipate heat into the surrounding air instead of the wearer.
Given a sizable force of lightly-armored troops bearing these shields, you just need to get the drop on him. Some stealth may be called for; you'd need to get the soldiers into the keep fast or quietly enough that the wizard has little time to prepare. Then, simply surround him, and an attack by him in any one direction can be met with a counterattack from the flanks.
Most magicians in fantasy have more than one magical power; the magician may be able to teleport, or cast illusory images, or may control more than one element. In many fantasy worlds that aren't games, wizards have few weaknesses against ordinary humans (except a legendary hero). You'll need to invent a few limitations of the arcane craft that can be exploited.
Do the attackers have to be good guys? There are a lot of ways to beat a fire-wielding mage, but most of them are against the rules in the good guy handbook.
Get some attack dogs and train them to charge across the castle gate. Put a harness on them and attach lots of dynamite or black-powder. Let them loose. If you're lucky, the mage will hear his guards under attack and try to help by setting one of the dogs on fire. BOOM! No more front gate or front gate guards.
...and that mage will be much more hesitant to throw fireballs in the future.
Now attach bags of black dirt to your soldiers' armor, and charge the hole where the gate used to be. Proceed with standard castle storming procedures from that point on.