It always used to be the case that you needed a licence to watch live TV or to watch recordings made from live TV. e.g. if a friend gave you a tape of a program that they had recorded, then you needed a licence to watch it (unlike hiring a film from a video shop).
This seems to have changed. The BBC now says that a licence is only required for viewing “at the same time (or virtually the same time) as it is being broadcast.” Catching up via iPlayer is specifically excluded and by clear and unquestionable implication (in my opinion), also in the case I outlined above.