I attended four different US undergrad schools, a US law school, and a Chinese economics school's master of laws program, and I have never heard of a greensheet. Like other commentators, however, I found that the Google God declares it is The Truth.
First, an Ngram:
I did not include before 1920 because there were zero uses before then.
Many colleges use a program called Blackboard, and Google Fu suggests that the "Syllabus Template" uses green text as a "placeholder" to tell the instructor to replace all of the green text. But, to jump from green text to greensheet is difficult to believe.
At Kutztown University of Pennsylvania, each instructor must give their syllabus to the administration and the syllabus must be color-coded based on department. Green is for the College of Business. I doubt that green is related to business schools, however, because most other uses do not seem connected to a department, see for example, Foothill Handbook. Nevertheless, it is possible that some long-forgotten bureaucratic decision requiring syllabi to be on green paper spawned this idiom.
It seems that greensheet is only used at a few institutions, so that makes it unlikely that some random person on Stack Exchange will know the answer. The Ngram suggests that the usage of the term is fairly new, however, so if someone were truly industrious, there are likely many people still alive who could help discover the origin.