If billionaires could suddenly gain immortality, would the world's governments finally act against climate change?

Yup. They would setup camps and gas the mortals to solve the gods' problems.

You wish.
If I were a sociopathic billionaire I would love some degree of global warming. The more you make part of the world unliveable, the more I can charge people for living conditions.
I can create bio domes that have clean cool air and charge people to live in there. I've now successfully monetized clean air.

Why?
They'd just need to get property in the safer places.
I think they'd improve research in automation and AI. So that they can have their stuff, without having to rely on regular people who'd be wiped out or affected by climate change in the long run.

I don't think so. Climate change is not a big threat for the rich. Actually they can even gain benefits from it, as it will push inequality to new heights.
- I don't think the billionaires' investments are going to be worth billions if the global economy collapses.
  
  I think relative values mean more than absolute for them. They just need to keep their position as the richest people in the world even in absolute sense they will lose. Money itself expresses relative values. So billionaires will continue be billionaires.
  
  We hoi polloi think in money numbers and what we can afford to purchase. For the Capital Class, it's all about power. Money is just how they keep score.
  
  Billionaires don't care about the economy. What matters is whom owns the means of production. And in today's age, whom owns the means of ideology production.
  Being g a saint to a group of bullies is worth everything.

At the current rate governments would spend the world's remaining time making a cozy place for them to spend eternity alone.

I'm pretty sure they would just become the goa'oul from Stargate and enslave the mere mortals.
- Time to kill some gods.
  
  Here, take a p90

Not at all. Those billionaires will still have massively huge egos that will prevent them from recognizing their own futures will be incredibly shitty.
They’ll use their wealth in the short term to build bunkers, etc. where they think they’ll be able to continue to live in luxury while the rest of the world burns around them. But no matter how good the bunker and how many supplies they squirrel away, they’ll eventually be forced to return to the real world, and won’t be prepared for the fact that their piles of money will be worthless if the planet is largely uninhabitable.

Then those billionaires would continue to keep hoarding wealth across the globe and let everyone else live in/near poverty. Even if they manage to curb climate change, the whole outlook would still look incredibly grim.

I don't know. I kindof suspect that:
The billionaires may somewhat believe their own propaganda and maybe the climate chang denying billionaires may outnumber the ones who are more in touch with reality.
The machines that capitalism has built to maintain and intensify wealth concentration may well have escaped the control of their creators. Corporations have wills of their own distinct from that of the people nominally "in charge" like the C-level leaders and board and shareholders.
Climate change itself may have already passed a point of no return or if it hasn't, it likely will before even the most powerful manage to redirect the momentum of the system in a different direction.

No.

Well, it's becoming a genuine concern because of the advances of medicine and the growing understanding of the aging process. Imagine a guy like Musk living for a 100 or 200 years, imagine the accumulation of wealth.
- That kind of understanding is centuries away from now. There is a lot of hype about medical junk, but it is just that. In the real world, the medical field is a commercial enterprise. It has a 3 sigma standard when all real science requires 5 sigma results to publish or make a claim. The vast majority of medical research is cherry picked which is absolutely unscientific. Go through a major traumatic injury like I have with my spine and you'll learn just how little humans actually know how to diagnosis and work on.
  Most of the hype is to get and justify grant proposals and to promote dubious commercial endeavors. We still can not even explain or synthesize life, and we've barely started to document a sizable chunk of our DNA. Like all the claims about fully mapping the human genome are crap. It is because they call a MASSIVE chunk of it junk DNA even when there is plenty of evidence otherwise.
  Biology is actually the ultimate technology. Once fully understood and mastered it will displace nearly every piece of industrially produced technology of our stone age of silicon. Such a future is inevitable, if we survive, because it is the only way to be long term sustainable and in balance with the environment. When that happens, it will be a time when people live the longest lives possible for a human.
  
  Of course it's not a 10 years from now thing, but it's probably going to happen someday, and a dude like Musk or Trump with an artificially extended lifespan could be absolutely devastating.

I'm surprised that no one has mentioned this is essentially the plot of fallout. The billionaires will solve climate change by culling the poor "destroying the world".

Lol nope.
Immortallity is not what drives billionaires.
One thing over everything else matters.
The highscore.
As long as it increases they don't care about anything else.

No, they’ll try to solve it using the free market and fail
- Nah, worse, they'll succeed from their perspective. At the expense of everyone else.

I think they'd also need to accept climate change exists

I think not, we would just get bigger boats at the sea.
And maybe they'd build some pyramids or something else crazy rich people do.

Even if they did, they'd never publicly admit it. There's far too much money invested in the industry.

No, it is not the individual action of billionaires but the profit driven nature of capitalism that is killing us. If billionaires were immortal they would simply move to more livable places, kill and impoverish large groups of us to reduce the environmental load or create ecospheres to house themselves while we burn.
The problem is not that all billionaires are bad people (they are), its that the economic system they uphold necessitates infinite growth on a finite planet. If these billionaires don't constantly grow their empires then a more evil capitalist willing to exploit others and the environment more efficiently will outcompete them and take their place.
Being shortsighted isn't inherent to a billionaire but it is inherent to the system that justifies their existence

If they're immortal, they'd live through climate change, so no, they wouldn't GAF.

You think Jimmy Carter would be 100years if he was a homeless person. The age-gap between rich and poor will only get bigger. Don't know if immortality is possible, but extending life is definitely possible. Extending life to 150 or even 200 is within our grasp, even today.
- Telomere therapies is but one path. There are many different paths that each contribute to a longer life in better shape. I've seen at least four different therapies, all very promising. I don't have article links at hand. Point being, that these therapies are so close that they might already be available, for the exploitation-class.

What makes you think they didn't invented immortality yet?
Isn't like they will be pandering it if they do.
Supposing they do I believe they will not give a flying fuck about climate change anyway.