I'm also getting the "Error 80" / "Err 80" error message. I bought my Canon 5D Mark III from my local camera store a few weeks ago. I had no problems at all for the first week or more. (Aside from the Error 80 problem, I love the camera so far and have no other complaints at all.)
I've probably taken 1,000 or so images so far with the camera. During the past 3-4 days, I've gotten the Err 80 (Error 80) message on the top LCD about 5 times. When the error is displayed, the camera ceases to function. The problem has always occurred immediately after shooting an image (and it appears that the last image does not get recorded). Each time, I have "solved" the problem by powering the camera off, removing the battery, re-inserting the battery, and turning the camera back on. The camera then seems to work normally for a while.
I have the camera set to record RAW images to the CF card (Sandisk 16GB "Extreme III" 30MB/s CF card) and small JPG's to the SD card (8GB Eye-Fi Pro X2 SD card). I have exclusively used the original/genuine Canon battery that came in the box. I have only used Canon lenses thus far. The camera has only been used in dry moderate weather conditions (50-80 degrees F, no rain, no sand, no humid environments, etc). Basically - I'm not doing anything unusual, and I'm familiar with Canon cameras (I've previously used 5D, 5D2, 1D4, 7D, and others).
I just called Canon CPS to ask them about the problem, and they said I was the first to call about it. They seemed to suspect the problem might be related to the Eye-Fi card. (They said the error 80 relates to writing the image to the card.) Canon suggested disabling the Eye-Fi card to see if that resolves the problem (as a way of ruling out if the EyeFi card is causing the problem). They were friendly and tried to be helpful and find a solution that didn't involve me having to send them my camera (which is appreciated).
Is anyone else who is seeing "error 80" using an Eye-Fi card? Has anyone found any solutions to this problem?