Not possible with real Linux, the CPU on the Furby is too limited while the smallest Linux out there still requires some more resources. Some small size Linux-like OSes do exist though: one commenter suggested Lunix (which I didn't know, thanks for the link), and a slightly bigger one is ELKS which runs on old MMU-less x86 CPUs. I managed to run it on a 8088 industrial PC ages ago.

https://github.com/jbruchon/elks

I should have a Furby buried somewhere; now that I think of it, it may be the right platform to stick a bigger brain into, make it wireless so that it could be connected to the home IoT network then signal events or alerts.