I think it definitely matters not to let it run after the yeast has consumed the sugars. Palmer, in How to Brew (p.122), tells us to take the starter off the stirer when it's halfway through, because the yeast needs to have the rest of the time to build energy stores (glycogen and trehalose). The implication there is that continued supply of oxygen hampers that.
I've mailed him about this, and didn't get a real answer. All he said was that he reckoned that at 12 hours after starting the starter would be about the right time to take the starter off the stirer. I can find no source to support the claim that you need to take it off - which of course does not mean it doesn't exist

. But I'd love to see it. (I assume that his source might be Chris White, as they're well aquainted, and White is on the list in Acknowledgements.)
Anyway, having yeast that has built good energy stores is vital. The yeast needs the glycogen for fueling the building of sterols and fatty acids, which are materials for the membranes. It does this using the oxygen we supply at the start of a brew, and the store of membrane materials is again vital for the yeasts capacity to multiply. The store will not be renewed through the brewing process (unless we supply more oxygen, which is what we do when brewing very strong beer), and if the synthesizing of membrane materials does not give the yeast maximum stores, it will not be able to produce enough new cells. The likely result is that we will have problems like incomplete fermentation and off flavors due to weak yeast not being able to "tidy up" waste products.
The trehalose stores are also important, as the yeast use trehalose when it is exposed to stress - and it is very likely to be exposed to stress.
I'm sceptical about the need for taking the yeast off the starter before all the sugars are used up. I mailed Jeff Caudill of Imperial yeast about this, and he said he saw no need for it, but that letting the starter spin after the sugars where consumed, would be harmful. So that's what I'm trying to live by theses days.
Now it's rather hard to decide when that crucial moment has come. There are lots of signs you can look for, but you can never be really sure unless you pour a sample and take a reading. So that's what I do. And I don't aim to wait as long as possible. I think it's better to get a little less yeast in order to secure maximum quality, so I tend to take the starter off a little earlier than I need to.
How much harm is done if the starter should be spinning an hour or two too long I do not know. Maybe it's not very critical? Or maybe it is? But as long as I don't know ... well, better safe than sorry.
I'm not a biologist or a chemist - I'm actually a retired history/litterature/philosphy teacher

- so reading scientific papers in the field is a little beyond me. I'm trying to piece together an understanding based on scraps and pieces of knowledge that i pick up, and I'd be extremely happy if anyone with some real (and I mean
really real!) knowledge could tell me what happens when the yeast is fed oxygen after it has eaten all the sugar. I know that it respires, and will use the oxygen to feed off the etanol, and if what Caudill said is right - and he ought to know these things - that process does not give us good yeast. But I'd really like to know
why.
Timing a starter is hard. You can't say how many hours it will take, because there are many factors influencing it. And so I sometimes miss.But if I do, and the yeast has been spinning too long, I do of course not discard the yeast. But I assume it has got somewhat depleted stores, so I then give it a vitality starter at the start of the brewday. That seems to work fine.