Life in Academia Would Be Hell
As someone who knows academia, I can tell you that an extended lifespan would make academia a heck of a lot worse. Academia right now works on a "dead man's shoes" principle. Career positions are so rare in many fields that getting a professorship position is often heavily dependent on the people ahead of you dying and opening up a position. If everyone lives to perfect health in their 200s now career opportunities no longer open up because older researchers just. won't. die.
This would also make getting a liberal arts degree or something similar an even worse career proposition than it is now. Normally, if you get a liberal arts degree or something else that has very little pragmatic value, there is always the potential of getting a job teaching in your field of choice. With people living to 200s, there is much less of a chance of that.
Tenure might also disappear as a concept. Tenure was mostly a mechanism such that older employees (who were assumed to be more knowledgeable about the workings of a university) could be more vocal with their criticisms or propose more controversial ideas and the universities couldn't just outright fire them just because they didn't like what they had to say. The assumption being that these professors wouldn't be around long anyway because they only had ten or twenty more years to live. But if people lived into their 200s, tenure suddenly becomes a lot more expensive (who wants to pay the salary of that one absentee professor for a century?) and colleges now have a lot of reason to abandon the practice.
This, in turn, would lead to an increase in stagnation in the sciences and humanities. Older academics that just. won't. die. will stymy scientific and social progress by impeding research that potentially threatens their pet hypotheses. Academics do this all the time today, by trashing grant proposals they don't like or giving nasty reviews to papers that disagree with them. That is if they don't just outright abuse their power by preventing people from conducting research that disagrees with them (I've seen this numerous times).
On top of that, radical scientific advances like evolution, catastrophic dinosaurian meteor extinction, the dino-bird connection, or plate tectonics only become widely accepted because all of their major opponents die. The initial opponents of the hypothesis become too emotionally invested in opposing the hypothesis and are unable to revise their worldview. Changing their opinion would be a loss of face, and they've invested too much to back out. Opponents of major hypotheses like Larry Martin and the dino-bird hypothesis or G. G. Simpson and plate tectonics never change their minds on the subject, they just died and opened the field to younger researchers who didn't have the emotional baggage associated with the debate and were able to look at things more objectively. If people lived to be 200 you would see tons of disproven hypotheses sticking around because some scientist is wedded to their pet theory, and you would get bunkum like orthogenesis sticking around to the present.
An Extreme Societal Shift Towards Conservativism
Older people tend to be more reactionary and prefer how things were "in the good old days". In general, across the world older people tend to vote for the more conservative/traditionalist political parties over more liberal/progressive ones. This kind of mindset seems to be increasingly adopted starting in what is now middle age when people have been around long enough for what their culture considered "hip" when they were young to be considered old and cliché by the younger generation (the "dang whippersnappers" effect). In modern society, this is offset by the fact that the old people tend to die, meaning politically younger voters and older voters are relatively balanced.
You've just set up a scenario where the old fogies outnumber the Young Turks by potentially as much as four to one. On top of that, they have an even louder voice because their numbers aren't thinned by dementia and other geriatric disease. As a result, in a democracy, the old timers are always going to win in political elections. Politicians also have much less reason to appeal to younger voters by including reformatory or progressive ideas in their political platform because the majority of their base are old people.
One might say that with an increase in lifespan the cutoff age for "old fogie" might just increase to seventy or eighty. But in reality all this might do is create age "strata" where certain age groups hold certain values, instead of an "old/young" divide. And the older age strata would still be more likely to be conservative, the shift seems to be relative to how many years lived and the passage of time more than anything else.
Immortal tyrants shouldn't really be a problem
Most modern democracies have laws in place that allow people to only rule for a few terms. Those that don't already have problems with leaders staying in power for long periods of time even without a 200 year lifespan.
