What happens when a professor can't fund his/her own lab? Given that funding rates are notoriously low, often in the single digits, what happens when a professor is unable to fund his or her own students? Given that TA positions are also limited, they may not always be able to get their students to be funded as a TA.

I am guessing that most of the time a professor only ever takes on a student if they think there is a good chance they can obtain sufficient funding to see them through. However, I am guessing the edge case of running out of funds has happened before.
If there is no money, then there is no money. What happens quite often is that some of her or his colleagues help out by hiring deserving students instead till extra funding has been found. This has the advantage that it diminished the risk everybody faces because of uncertain funding, but it requires quite a bit of trust and good will in the department. So the worst case scenario, that the professor has no money and nobody wants to help her or him out, can definitely happen. 

This also depends on the contract you have. In my case I got a 5 year contract to do my dissertation and some teaching. In that case, my position was safe: if somehow the professor's funding somehow failed the university had to come up with the difference, since the contract was not between the professor's lab and me, but between the university and me. This is in line with @IanSudbery s comment that, however much professors think of "their" lab as her or his private kingdom, it is only part of the university and not a separate entitiy. The fact that the university is finally responsible for paying the wages meant that the university required my professor to have guarantees for the funding for those five years before agreeing on the contract.