How much of a role tuition revenue plays in underwriting research? I knew that TA positions in American schools are usually borne out of tuition revenue. Tuition revenue (that is, the cash that comes from the sometimes exorbitant undergraduate tuition) is usually unrestricted so one would wonder whether tuition money is directly used in order to fund research.

But other than start-up research funds, what other research expenditures are borne out of tuition money, if any? Would a university use tuition revenue to make up for a shortfall in research grants?
Much of my answer here, holds in this case. It's basically impossible to tell. If a new professor gets a lighter teaching load in their first year, is that "underwriting research" even if the professor's nominal work distribution is 1/3rd teaching, 1/3rd research, and 1/3rd service/admin?  

Professors are paid to teach and research and serve internal university goals. How much they do each of those day to day is hard to tell. Now, that being said, if a professor fails to bring in grants to cover their summer months, I doubt many universities would indefinitely provide for them out of general funds.