Does undergraduate tuition help fund professor salaries? Does it help fund graduate student salaries, especially if the grad students are TAs? What about lab facilities?
(This is very much a US centric answer; other countries structure their university systems differently.)
This is a difficult question to answer because money is fungible, so in some sense it doesn't matter what the nominal answer is---if the money isn't going to fund those things, it's filling in other expenses so that other sources of revenue pay for those things. But in practice, most universities put undergraduate tuition directly in their "General Fund"---a big pot of money, which is then divvied up to support the general operation of the university, which includes all the things you mention. (Note that many sources of funding for the university are dedicated for particular things, and don't go into the General Fund, but undergraduate tuition usually isn't one of them.)
The University of Michigan has a nice online presentation about this, and as far as I know, that's typical: undergraduate tuition pays into the general fund, which pays for most of the university's core activities.