whether necessary to continue to work on unfinished academic work When one finishes a degree or a research position, the boss asks to carry on sth unfinished (unpaid), does one need to agree to do so?
No, you don't need to agree to do it at that point, unless you already have agreed or committed previously. However, for your future career it would be good to give it some consideration, at least, and to work in in if possible. 

I'm assuming that "the boss" is a professor, actually, who can aid you in the future. 

But "unpaid" is a big issue. If you can get something back for your efforts either hard currency or academic currency it would be a good thing also.