What percentage of a professor's salary is paid for by tuition?
I think that you'll never be able to sort this out. Most universities cross-subsidize departments to the point that even individual departments don't know how much of their own faculty lines come from the tuition of students majoring in that subject vs. tuition from other departments' students. Does the English department fully fund itself through the fees students pay for English classes? I doubt it. State funding, though a diminishing portion of most state universities' budgets, picks up some fraction of the costs. 

Also, departments see level funding in the face of modest fluctuations in class enrollments. If 45 students took Calculus I last fall and 52 sign up this fall, the department doesn't see increased revenue for that. The department has a number of faculty lines coming from the dean's office to teach the projected enrollment. If that spikes or drops off substantially such that more instructors are required, then the department has to negotiate more or fewer lines with their dean. 

That being said, I think that most departments negotiate expected breakdowns in effort with the faculty they hire. Of a typical 9 month appointment, a faculty member might be expected to do 35% teaching, 35%  service, and 30% research (or whatever), and that might be the expectation regardless of whether the teaching load is a 1-1, 2-1, or 2-2. Where, again, it's entirely unclear whether that 35% teaching is completely funded by tuition or partly funded by tuition, central endowment, and/or a local named chair.