In 1984 a Festschrift was published in his honour, Studies in Seventeenth-Century English Literature, History and Bibliography, edited by G.A.M.
Birrell was born in London to a Scottish Presbyterian father and an Irish Catholic mother.
He served in the Royal Armoured Corps, in the Netherlands and in India, returning to Cambridge in 1946 and graduating with honours in 1947.
Birrell is best known for his 1986 Panizzi lectures, published in 1987 as English Monarchs and Their Books: From Henry VII to Charles II, a groundbreaking study of England's Royal Library.
A collected volume of his articles was published by Ashgate in 2013 as Aspects of Book Culture in Early Modern England, edited by Jos Blom.