So at the very least, the origin of this story predates 1991 by at least two years.
I don't recall the name of the building but here it is on Google maps.
https://www.google.com/maps/@40.106201,-88.2268272,3a,75y,91...
Edit: It's not clear from my original comment but the reason for collapse would presumably be a nuclear strike. I remember this because this was a time when we grew up with a constant fear of a Russian nuclear strike and I couldn't help but wonder why anyone on earth would want to nuke Champaign.
Edit: Ah, here we go! It is the Foreign Languages Building (FLB), later renamed. I remember having to trudge here at 7ams on snowy winter days to listen to Japanese language cassettes.
https://uihistories.library.illinois.edu/virtualtour/maincam...
Edit: And here's a contemporary article about the FLB, which also cited some of the crazy rumors about this building.