A DAMNED SERIOUS BUSINESS: My Life in Comedy by Rex Harrison (Bantam Books: $21.95; 288 pp.). The complete quote from which Rex Harrison has taken the title of his autobiography is David Garrick's famous avowal that "Any fool can play Tragedy, but Comedy, Sir, is a damned serious business." It's a prescription with which Harrison agrees--to a fault, in fact, for the actor reveals little about himself in this volume other than his constant need to work. Harrison began his theatrical career in his early teens in Liverpool, and after a few years of knocking around in repertory and touring companies made a name for himself in higher-class West End comedies. In the 1930s, Harrison began doing film work as well, and by 1941 had acted in the Noel Coward and G. B. Shaw plays that would forever define his career. Harrison, who died last year at 82, comes through as the last of a once-dominant species; it's difficult to imagine another actor today in constant argument with a director over fidelity to a playwright's script, as Harrison was with Moss Hart during the translation of Shaw's "Pygmalion" into "My Fair Lady."