to run a car engine so fast that one spins the tires so that rubber is left on the street. (See also lay (some) rubber.) : When George was at the age when the greatest thrill was burning rubber, he began to shave once a week. Dictionary of American Slang and Colloquial Expressions by Richard A. Spears.Fourth Edition
Drive very fast, as in We'll have to burn rubber to get there in time. In this bit of automotive jargon, the rubber refers to tires that heat up when they rotate suddenly at great speed. [Slang; mid-1900s] The American Heritage® Dictionary of Idioms by Christine Ammer.