This is the classic "without death, authoritarian depots will be in power forever".
But it's actually the wrong way around. Hope that the despot will soon die saps peoples' will to do the difficult and dangerous job of removing them. Take away that hope and they are forced to find the courage.
People might be willing to give their known-to-be-finite lives in pursuit of the Greater Good, but the equation changes greatly when you're giving up an eternity for the chance of making things better. More of the population would be aged too, so I think it'd be a lot harder to find the kids that usually die in wars.
Especially true if it is the AIs that get us the immortality- it definitely won't be equally spread, and any incumbents have a massive advantage.
I think more people would settle for worse conditions to stay alive. Do you think you'd be more likely to revolt at 150yrs old when all your family and yourself can live extremely long to forever? Or do you think the threat of death by killing wouldn't be an issue?
I don't think anybody wants to live forever in an old person's body. Frankly, I'm not interested in extending my lifespan unless I can stay young or regain youth.
Also, if you eliminate all the aging-related causes of death the average lifespan is still only like 1000 years or something. Nobody's getting eternity. And yeah, for the chance to make a big enough change of the right kind to society I would indeed give up a 1000-year lifespan. My memes have fully subjugated my genes, and I'm at peace with that.
Forced to find the courage or die trying to. And I really doubt that knowledge that their despot will eventually die has much effect on people's motivation to rebel as in most cases there is a clear line of succession. In many cases there isn't even a single despot to point to. This isn't even considering how such anti-aging technology would prevent despots' mental acuity from deteriorating as they aged - in fact they would probably grow sharper as they gained more experience so to speak. And aging/succession have almost always been one of the greatest destabilizers of a successful authoritarian regime.
But it's actually the wrong way around. Hope that the despot will soon die saps peoples' will to do the difficult and dangerous job of removing them. Take away that hope and they are forced to find the courage.