The Vatican is letting him stay there until Mater Ecclesiae is finished, which is only a few months away.
I doubt very much the new Pope will throw him out in the meantime.
After seeing him today, we can’t even be sure he’ll live long enough to move into Mater Ecclesiae...
Okay, so he’ll be there for a few months. That’s not like living there permanently. Oh, and I doubt the future pope would kick him out. But then again, you never know how a future pope might react to having a retired pope around. The last time it happened, it was a different world. Some of the potential popes may not like that idea.