Weird, So you are using the green blue red output composite from the mini to the DVD for video or the yellow RCA video line? Kinda sounds like HDCP but more like a inactivity timeout were the Mini is detecting the TV is off and it's misinterpreting the down HDMI signal as time to shut down?? How are you sending sound to the DVD player, component RCA or does the DVD have Toslink input?
Perhaps other's will voice in on how the Mini handles inactivity as there is a certain amount of time that passes with no activity from the remote were the Mini will release the tuner on the host but I'm not sure if that's related or not.
That splitter I referred might fix it as it would maintain the HDMI output signal regardless if the Mini times out if the Mini is somehow monitoring the status of the HDMI status of the TV but that is just a guess. I don't suppose the DVD player has HDMI input by any chance?
Sounds like you are encountering the timeout "feature" perhaps?
If it's the Mini's timeout doing this, it could be as simple as hitting the pause and play button the remote before the timeout expires (90 minutes??).