It's not the existence of books that have been made illegal, but those that have been deemed offensive and/or against the State in some way. This inevitably had devolved into any book that promoted independent thought, poetry, beauty, and The Bible itself. Because any set of criteria used to ban books would be arbitrary in nature, sooner or later almost all books would be banned, and the government would become dependent on the firemen and the ban to maintain control over the populace.
Not all books are illegal and yes people do read in this society. The firemen burn thousands of titles that have been banned because they go against the doctrines of the government of this society. A book is banned and burned if it encourages free thought and open mindedness- you can only imagine the titles being burned! There are even books which have familiar titles, but have been rewritten to reflect the ideals of the society so that they are unrecognizable to any reader who may have read that particular title prior to it's new edition release.
No, the fireproof houses themselves do not burn, but everything inside them can burn because not everything is fireproof.