Born in Leicester, Gurr was raised in New Zealand, and educated at the University of Auckland and at Cambridge University. He has taught at the Universities of Wellington, Leeds, and Nairobi (1969—73); at the latter institution he was also head of his department. He taught at the University of Reading before his retirement.

Gurr co-wrote a 1981 study of Katherine Mansfield (with Claire Hanson) and two books on African literature; but he is best known for his books on Shakespeare and his contemporaries, and the theatre of that historical era— books that are recognized and utilized as essential references on English Renaissance drama. He has authored a wide range of articles for both scholarly journals and general-interest periodicals, and has edited several of Shakespeare's plays and several plays in the John Fletcher canon. He was chief academic advisor to the project to rebuild the Globe Theatre in London and also advised on the construction of the Blackfriars Playhouse in Staunton, Virginia, where he has also lectured at the Blackfriars Conference and for Mary Baldwin College's M.Litt/MFA Program. For ten years (1988—98), Gurr was the editor of the Modern Language Review.