I just had the same problem this morning - blue screen when I try to play a recorded tape, and was able to fix it (it seems). It may have been caused by my trying to control the camera through a DV connection with a Sony DVD recorder. Essentially, I fiddled with it.
I tried all the suggestions in the manual (reset the machine, take the battery out, etc.) but nothing worked. The machine was acting as if it couldn't read the tape (no date symbol), so perhaps it was not engaged. Finally, I opened the tape eject door WHILE the tape was in play mode. The tape ejected, but then when I reinserted the tape, the machine engaged the tape again and started to play. Maybe it was just luck, but I hope it works for you and, seemingly, many others who have had this problem. Other than this, I've generally been very happy with the camera - while a bit bigger than the latest models, it's easy to hold, and seems to have all the right features.