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.

To the best of my knowledge, my university salary is paid out of what the university calls the "general fund". That fund includes the money the university gets from tuition, the money it gets from the state (except possibly for state money earmarked for specific projects, like a new building), and I think also the indirect cost funds from research grants (and maybe other income sources too). Once that money is in the general fund, it gets completely mixed together, regardless of source. And I get the same salary, entirely from that fund, whether I'm teaching my normal course load or whether part of my time is devoted to administrative duties, or whether I'm on sabbatical. So, as far as I can tell, the proportion of my salary that comes from tuition is just the proportion of the general fund that comes from tuition; no finer analysis is possible. (Unfortunately, I don't know what proportion of the general fund comes from tuition, but, since the University of Michigan is a public institution, that information is undoubtedly publicly available somewhere.) [Edit: Deleted the sentence about "25% quite some years ago but not from an authoritative source". That 25% was for state support, which has probably decreased since then. Tuition is probably a much larger share of the general fund.]

I agree with Bill's answer, but also want to contribute a few more points:

• In many other European countries, students pay no tuition, and thus there can be no contribution.
• Faculty in the US can be paid according to either a 9-month or 12-month salary scheme. When they're paid for nine months per year, they are expected to raise their remaining salary through external grants. Then you have to ask the question of which salary you're referring to in your calculations.
• Salary differs widely among faculty members at differing levels of pay.
• Graduate tuition is often an internal accounting device, as graduate students themselves (and particularly PhD students) are rarely expected to cover their own tuition costs.