Publishing previous results from a previous lab
I think that so long as the principal investigator is actively involved in the planning, performance, or analysis of the research being funded, it is the duty of any researcher working in that group to determine the PI's status as a co-author. However, only if there is no active intellectual activity taking place—in other words, it's an entirely self-driven initiative, then it might be possible to say that the PI doesn't merit co-author status. (Even then, though, the provision of financial support should be clearly recognized.)