The book would not be automatically removed. It would be your responsibility to remove it once you were done (excepting Mobipocket format books, which would expire on time if only tweaked and not stripped).
Getting library books onto your Kindle is possible, but you have to be willing and able to break and strip the DRM on most of them, and also converting ePubs to Kindle format.
The first part is technically illegal in a number of countries, and some people are not comfortable with this method and prefer to buy a second e-Pub supporting reader instead if they can afford one, or just do without library downloads altogether.
Others consider it fair use as long as their tax dollars paid for the library service and they don't pass the files around and delete them off the Kindle once they're done. It's really up to the individual to decide.
Hope this helps.
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