When I press the depth-of-field preview button on my 5D3, an Error40 message appears on the screen. I've tried with and without different lenses, remapping the button to every other function and even turning off the button, and I still get the Error40 message in every scenario. I've tried with multiple OEM batteries. Has anyone else experienced this issue?
From my own research, the Error 40 is an electrical issue. The easiest thing is if it is related to a battery error. The best thing to do is replace the battery with a known good battery, preferable an Canon original one and try to turn on the camera again.
Another possibility is that moisture has ingressed into the button seating when it was being pressed - from what you write, I suspect you use the DoF button fairly often(?). That should not happen as the body is moisture-resistant.