You can’t (and shouldn’t) keep a library book forever, but if you’re not quite done with a book, then you can do the following to finish it:
Do not close the ebook to look at another book or adjust a setting. Rather than powering off your Nook, only let it sleep. Do not hook the Nook up to the computer or otherwise charge it.
According to other Nook owners, this will keep the book from expiring. Of course, you’ll eventually have to charge it, but if you’re lucky you should be able to get it done before you’re done reading. A better idea is to always check out your Nook ebooks for 21 days, since you can return it if you finish early.