As best I can tell, there are four people named in the song We Didn’t Start the Fire who are still alive (counting either people mentioned by name, or people uniquely personally identified, ie, QE2 in the lyric “England’s got a new queen”). Without looking up the song lyrics, can dopers name all four?
Tiger Woods, Obama, Taylor Swift, and Robert Downey Jr.
(OP didn’t specify which version of the song we were to use.) 
Only one I could come up with was Bernie Goetz.
Bernie Goetz is correct.
Besides Goetz, I’ve got Dylan and Chubby Checker. Can’t think of the fourth.
Damn, I forgot Chubby Checker was in the song and I knew he was still alive!
Brigitte Bardot is still around.
How about the “children of thalidomide”?
I think the argument there would be that they weren’t really “named”.
Also there are still Ayatollahs in Iran (although not that one) and presumably Russians in Afghanistan.
Going through the lyrics, and people who are specifically named, it looks like Goetz, Bardot, Dylan, and Checker are the only ones still alive.
Two other arguable corner-cases:
- “British Beatlemania”: Two of the four Beatles (McCartney and Starr) are still alive.
- “Wheel of Fortune”: Both of the hosts of the show in 1989 – Pat Sajak and Vanna White – are still alive, and the show is still in production.
All four have been named:
Bob Dylan, Bernie Goetz, Chubby Checker and Brigitte Bardot.
Time to play “One of these is not like the other …” BB is a woman.
Is that Bernhard Goetz, the subway shooter?
It is.
Wow! Shoot a guy in the spine, end up immortalized.
Adding to this, the OG members of Sugar Ray are still alive: Mark McGrath, Rodney Sheppard, Murphy Karges, and Stan Frazier.
Two of the three founding members of Alabama are still alive: Randy Owen and Teddy Gentry.
All four original and only members of U2 are still alive.
I’m deeply embarrassed to admit that in my initial mental playback of the lyrics I initially thought, “Oh, Sugar Ray Leonard is still alive” only to quickly realize the song reference is to Sugar Ray Robinson.
Assuming that I’m not being whooshed: I don’t think that any of those mentions in the song refer to those bands, but to boxer Sugar Ray Robinson, the state of Alabama, and the U-2 spy plane.
I wanted to make that U2 joke, but just couldn’t come up with a version that worked.
I will add that the spy plane is still in service for a little while longer:
While not directly named, Caril Ann Fugate who accompanied Charles Starkweather during his murder spree is still alive.