I had the LENS CAP error on my JVC GR-D70 camera. I now solved this myself and I like to share my experiences for your benefit. After some google-ing I found that this is a common problem for just out of warrantee JVC camera's. It also seems to originate in the CCD part. JVC repairs this problem by replacing the CCD which seems to cost about half the new price of the camera. Then I found this:
Here you can find the service manual for several JVC cameras available for download. I studied the manual for my camera and came to this conclusion:
The CCD chip is soldered onto some flexible PCB that doubles as a flat wire connector to the main PCB. Soldering the 14 solder joints form part of the assembly process of the camera. This means the connections are manually soldered and therefore probably a weak point in the camera's circuitry. I then took apart the camera by following the clear service instructions from the manual and located the CCD part under suspicion. On my model, you'll find it right behind the viewfinder part, under a sticky isolation pad. The 14 solder joints are easily accessible. I carefully resoldered all connections, double checked that I made no shorts between adjacent joints and put the camera back together. I put the battery back in place switched on and to my pleasant surprise, there was a perfectly fine image!
For completeness: I also unplugged the connector on the main PCB side of the CCD wire and reinserted. This could be enough to get it going again, you may try this first. A second problem was the viewfinder that got stuck inside after my 3 year old son gave it a smack. By following the disassembly notes in the service manual I could find and repair the cause of this mechanical problem rather easily.
Concluding, the CCD wire connection CN1b, Fig. D1 and CCD BAORD ASSEMBLY Fig.2-6-1 seem to be badly connected and cause the "LENS CAP" problem with these cameras. Since there is no CCD signal available to the camara, the software concludes the lens cap must be still on. Effectively, other than the actual live image, everything else is working fine with a camera in this condition. In my case, resoldering the bad connections fixed the problem (for now). But be careful: only do this if you are experienced and feel confident with soldering small scale electronics. The connections are really small and close together, it is hard to see by naked eye what you are doing. Double check for short circuiting the connections before you put the camera back together or you may create further damage.
Maybe this can help somebody fix this problem like I did.