I'm curious why he thought putting BS-CS on his resume was a good idea in the first place. Alidan is very right about his faked degree being almost completely inconsequential to his daily work.
Does Yahoo! have some kind of bizarre prerequisite for employment that all employees have a CS degree? Even for HR, Marketing, Sales, Administration, etc. employees? If so, that sounds kind of like a "You have to walk a mile in my shoes before I'll let you join my company" kind of thing. Not all bad, because engineers of all flavors tend to be underappreciated in their jobs. But I think that if it's true he's written substantial amounts of code - a possibility that could be vetted with his alma mater - then that should be enough miles walked to fulfill such a requirement, especially since his school didn't offer the option of a CS degree.
So why the fake? I suspect that a) there was some kind of such a requirement for employment, b) Hart knew that Thompson's education background wouldn't suffice, c) the two of them decided to fake it to get past the cultural issue of having employees not recognize Thompson as "one of them", but Hart felt that Thompson was otherwise fully qualified for the job, and d) neither of them ever dreamed that someone would research Thompson's background in such depth. But clearly neither one considered the SEC filings and the increased scrutiny that such documents attract. Which would be foolish, since $billions are at stake.
Anyway, that's my theory. Thompson and Hart tried to take a shortcut to get an otherwise qualified candidate in the door, and respected by his employees. But it backfired, and now this group wants to leverage that to capture the board. If anything, I would trust him more than them.