I’d like to bump this suggestion since I’ve wanted this for some time now.
When I log in to play, I don’t like concentrating on dailies. (Doing so leads to boredom and frustration for me.) I would like to complete them every now and then, however, but during the normal course of gaming.
I don’t see the harm in allowing partially completed dailies to carry over so long as you can only earn one daily achievement reward per day.