I've heard that, over the course of the earth's history, climate change has occurred a number of times for various natural reasons, and even today part of the cause of what could be (I don't want a debate on whether it really is!) happening in our real world could have natural contributing factors. So what would be some good fantasy reasons for a TTRPG campaign world to undergo climate change?

The standard device would be angry gods, particularly gods of weather or nature. There could be an extremely powerful former dragon turned demilich lying deep within the heart of some mountain, unintentionally, unawares, and uninterestedly messing up the world's climates with its own incomprehensible magical investigations. The ruling wizards of a land such as the Forgotten Realms Thay could be working powerful weather magic for their own geopolitical ends. Or, as was alluded to near the start of The Malloreon by David Eddings (when Garion causes a sudden thunderstorm to appear from clear skies for political purposes), some narrow-visioned interference with local weather somewhere in the world could be having a ripple effect throughout the entire world's weather patterns.

Anyone else have ideas?

Similar reasons to real climate changes.  Just, with magical/fantastic effects triggering them.

Short Term: Particularly Large Volcanic Eruptions
            Solar Eclipses (more so with magic that might extend the eclipse)
            Wildfire Seasons (that are particularly harsh)
            Meteor Strikes

Long Term: Massive deforestation
           Massive industrial output (forges, chimneys, artificer labs, etc)

Also, much like your suggestions, a city-state or region of mages that constantly alter the weather for their own region might have odd effects either continent- or world-wide.  (Mercedes Lackey tends to include magic that affects weather in this way in her series, including the Obsidian Trilogy and the Elvenbane series.)  It doesn't need to include a heavy-handed chaos-theory fueled 'every small change affects everything' in the way David Eddings used it.  Careless, negligent, or cruel mage-governed cultures can mess with the world intentionally or unintentionally.  And, it would let a clever GM have the climate affected more regionally than globally if they want.

In reply to Godzfirefly (msg # 2):

I can see why limiting the broader effects of a local weather change, or a bunch of local weather changes, would make sense; but why do you call it "heavy-handed?"

In reply to mickey65 (msg # 3):

Eh...that may be a bias against the way David Eddings handled it showing up in the way I write about it.

I guess I'm saying it's heavy-handed because it felt used repeatedly as an example of why the characters couldn't just do anything they want, without real variation in describing it.

Also, there was never a real explanation of why it's okay to unilaterally alter the weather in one case but not another.  Just "I know what I'm doing," without showing us what it means to know what you're doing.

That was sort of the core of Belgarath's character.  Despite it being highly frustrating by the end of the series.

The ice giants of the far north are marching to war against the fire giants of the equator, bringing their slow-rolling but implacable glacial forts with them. At the same time, the rain giants of the peaks are starting trouble with the sea giants. Each race of giantkind brings its own elemental climate system with them as they go to battle, creating weather chaos - and the victor of the wars will inadvertently shape the climate of the epoch to come. Smaller races may ally themselves with different giants depending on their own environmental preferences; the lizardfolk look forward to a balmy warm period if the fire giants emerge victorious, for example.

Energy cannot be created or destroyed; even magic cannot escape this iron law. Every spell that is cast creates a net loss of order and energy in the world, due to entropy and inefficiency. The effect of hundreds of even minor spells cast every day around the globe is pushing the world towards an ice age. (This is basically A Wizard Did It, but with the analogy being closer to small-scale carbon emissions mucking up the atmosphere than an arcane nuclear war).

The sun is the egg of a cosmic phoenix. It is about to hatch, and its fiery wings will bathe the world in solar flares. Worst case scenario is boiling seas - followed by eternal cold and darkness as the stellar firebird flies away. More mild effects might be increased wild magic / arcane chaos as the solar flares interfere with magic (ala coronal mass ejections vs. our world's satellites and power grids), droughts and heatwaves, sunburn intense enough to wound or kill.

In reply to Godzfirefly (msg # 4):

Not to go too far off-topic before I bring it back on-topic again, but to me it was a way of dealing with the fact that Garion COULD indeed do absolutely anything. He was theoretically omnipotent, and limits have to be imposed on an omnipotent character if you want him to be the protag of the kind of story Eddings was writing. Without such limits Garion would have seemed profoundly unimaginative and stupid in doing things like traveling across the entire Continent of Mallorea slowly, on horseback. I agree that there could have been at least some thinking behind the limits, but back then fantasy writers often blew off complexities with New Age tropes.

Back on topic. My personal taste is for things in a campaign world to be consistent with each other, often to an extent that players never become aware of. If the game world makes sense to me, there's a much higher chance it'll make sense to the players. That's why I said what I said about it making sense to limit the global implications of local weather magic. If local weather magic affects the whole world, then every action a PC takes does too, and that's a nightmare for me and for the players. So I thought you might be saying something like that, but I now know what you were indeed saying, so we're good. :)

Any magickal working strong enough to affect a set of global weather patterns would be high magick, indeed.  Whether the resulting alterations were for good or ill is another very subtle matter.

Anything (magickal or non-magickal) which cast large clouds of smoke, ash, volcanic dust or ash, or other sky-blotting material will certainly have an effect on at least a regional weather pattern.  The same for any phenomenon that causes a great discharge of water vapor.  The former will tend to cool the affected area, the latter will tend to warm it up.

I've heard of one story where an ancient rave of lizard people created artifacts that would generate massive amounts of heat so that they could reverse an ice age in their home planet. The plan for our own hold by a long war but when the ear was done they tried to activate the artifacts.

This could be used a number of ways. Any species could magically terraform a planet to be more habitable for themselves at the expense of other races. A furry race could be coming the planet or a lizard race could heat it up. If you want they could be stealing energy from one world and moving it to another (leading too confusing mysteries like "why are lizards and fire people making the world colder?") It could be a cruel race that doesn't care if they kill off others or a desperate race who's homeworld is dying (so after you defeat then you have to take on the species that made then run for their lives and is now it to conquer your homeworld).

Climate change and sorcery?

Watch Fire and Ice (if you haven't already). Art by Frazetta. IIRC it can be found in full on youtube.

https://en.wikipedia.org/wiki/Fire_and_Ice_(1983_film)

In reply to Morty (msg # 10):

The Dark Sun world...

A group of dwarves burrows too deep into a volcano, triggering a series of eruptions that kick clouds of ash into the sky and blot out the sun for years...the region around the volcano grows hotter as lava flows continue to expand through the network of mining caverns.

Someone already mentioned dragons.  I also like the notion of warring giant or elemental factions.  Some wizard was playing with spells he couldn't fully control and changed the whole climate (you could even have this result in some cataclysmic tectonic/seismic events, where parts of the world actually shift position and old maps become useless...)  Some mad mage calls meteors down ftom the heavrns to wreak vengeance on an opponent.

A cataclysm,  much like what happened on Krynn, caused areas to shift.  Great harbors became land locked deserts. A great human nation was swallowed by swamps. On the other side of the world a great lava lake appeared spewing black ash and glass everywhere.

All because the gods got angry