The Mr. Burbage who is mentioned a few times in the narrative is a real, historical character. Richard Burbage was a leading actor in Shakespeare's acting troupe at the Globe Theatre in London during the Elizabethan era. Burbage was born in 1568, and began his career at the Globe somewhere around the year 1584; he performed until the time of his death in 1619.
Burbage played a number of lead roles in Shakespeare's plays, including Hamlet, which is the work young Widge is commissioned to steal in The Shakespeare Stealer. Mr. Burbage is mentioned only a few times in the narrative, but from the references to him, it is clear that he is an illustrious character. In Chapter 14, Sander and Widge are whitewashing the thatch on the theatre's roof when Widge drops a paintbrush. Sander expresses the hope that it was not Mr. Burbage who was struck by the brush, but to his dismay, it is even worse than that - the brush had landed on Mr. Shakespeare himself! In Chapter 17, Julian makes a comment about becoming "as famous as Burbage," and in Chapter 23, Widge mentions that he had had the privilege of sharing the stage with "our Hamlet, Mr. Burbage," whom he found to be "patient enough," but a man who "seldom displayed any real warmth or friendliness."