Short answer:
Written works weren't ordinarily checked out, they were read on premises. But if you were important enough, you could borrow it . . . and if you were that important, due dates likely didn't matter.
Discussion:
There's a term that now sounds a bit quaint -- "lending library"-- and the lending was at one time a necessary qualification, because the idea was novel. The essence of a library had been a place where one could _read_ books, but taking one home with you is a much more recent phenomenon, becoming familiar well after printing technology lead to the proliferation and replaceability of printed books. No one's going to lend you a one of a kind: the Lindisfarne Gospels may be at the British Library, but that doesn't mean you're going home with it.
The lending library as an idea develops in 18th and 19th century Britain, with a number of associated terms, you'll also hear "circulating library", "subscription library" or "commercial library". These organizations were originally conceived somewhat like book clubs, a way that men and women (and there's a substantially gendered context for these institutions) of ordinary means might have access to the wealth of printed information being produced, albeit often expensively in relatively short runs. The fully stocked bookseller was "not yet a thing"; one typically bought newly published works from a publisher/bookseller, which might subsequently become hard to find.
Libraries in the ancient world start out as personal collections -- access to which was entirely up to the owner of the library, typically a wealthy man, or perhaps an institution like a temple. It's in the Roman Empire that we get the notion of a library open to (at least some of the) public.
[Edit] I happened on a specific instance of Romans describing borrowing from a library:
The domus Tiberiana library figures in an episode involving Gellius's friend Cornelius Fronto. The future emperor Marcus Aurelius writes to Fronto that he has just read two orations of Cato the Elder. He anticipates that Fronto will send a slave to the Palatine library to get the books for himself, but warns Fronto that he has taken these books from the Palatine library with him to some residence outside Rome.Therefore, he says, Fronto must "work on" the librarian of the domus Tiberiana and bestow something on that librarian to obtain another copy of the orations. That an imperial heir could "borrow" books from the libraries should be no surprise; that even a Fronto had to go hat in hand to a librarian for the same privilege might also be no surprise, for ancient librarians may have been just as wary of their patrons as are some of their modern counterparts.
That's a wonderful bit of detail but we don't have any indication that their materials were borrowed as a matter of policy, but apparently influential individuals may have done so on occasion. When it comes to the public libraries, it appears that quite a lot of policy was up to the discretion of the librarian, and if your name is Marcus Aurelius there's no "due date". On the other hand, I suspect that if Marcus Aurelius wants it, and you borrowed it, its due yesterday. The impression left isn't so much of an organized system as of grad students sharing some rare unpublished materials in a collegial fashion.
The historian and geographer Strabo tells us that "Aristotle was the first ... to collect books, and he taught the kings in Egypt how to organize a library", and while that might not be exactly true, it seems to have been a common perception in the ancient world, where it was understood that book collecting started in Greece, developed in Egypt, and then became popular in Rome. Note that Strabo doesn't mention other, older, near Eastern archives of cuneiform tablets; he's talking about what he knows of the origins of the traditions he's familiar with.
Some of our best evidence for how Romans stored their materials -- scrolls, not books-- comes from the "Herculaneum Papyri", carbonized scrolls that were preserved in the Villa of the Papyri by the eruption of Mt Vesuvius. This villa, the home of a wealthy man -- some speculate Calpurnius Piso-- had an extensive library as well as many works of art. We have many subsequent examples into the contemporary era of wealthy men with private libraries allowing scholars to consult their materials, but this would likely be on site, certainly we have no indication of lending as an established policy, though it might happen ad hoc. Libraries could also be attached to temples, as for example Eumenes' library at Pergamum was attached to a shrine to Athena.
We're told that Julius Caesar intended to establish two public libraries in Rome -- one of Greek texts, the other for Latin (this segregation by language seems to have been the usual practice). After his death, the Emperors Augustus and Trajan both built public libraries, more or less in accordance with Caesar's project.
Sources:
Allan, David. "A Nation of Readers: The Lending Library in Georgian England", (The British Library, 2008)
Sider, Sandra. “Herculaneum's Library in 79 A.D.: The Villa of the Papyri.” Libraries & Culture, vol. 25, no. 4, 1990, pp. 534–542.
Casson, Lionel. "Libraries in the Ancient World", (Yale University Press:2002)
HENDRICKSON, THOMAS. “The Invention of the Greek Library.” Transactions of the American Philological Association (1974-2014), vol. 144, no. 2, 2014, pp. 371–413.
Molina, Manuel. "Archives and Bookkeeping in Southern Mesopotamia during the Ur III period", Comptabilités [En ligne], 8 | 2016, mis en ligne le 20 juin 2016
Dix, T. Keith. “‘Public Libraries’ in Ancient Rome: Ideology and Reality.” Libraries & Culture, vol. 29, no. 3, 1994, pp. 282–296.
Matthew Nicholls, now Classics Professor at University of Reading did his 2005 Oxford doctoral thesis (Americans would say "dissertation") on "Roman Public Libraries", useful reading.
Houston, George. "Inside Roman Libraries: Book Collections and Their Management in Antiquity". (University of North Carolina Press, 2015) [I've only just found this book, can't claim to have done more than skim it, but seemingly comprehensive]