Tv on is not necessary although if you have a cable box and need the DVR to change channels using the cable box, then you will need to leave the box on. Generally, with a DVR, its not so much about recording a certain time period as it is about recording a show (i.e. 24, Lost, whatever). At least with the Replays and TiVos. It will catch that show whenever it is aired, even if its on at a "special time" (network lingo).
EDIT: Of course, you can set it to manually record a certain time period if you want, just like a VCR.