The TV show featuring the White Clown is meant to show the society of people in "Fahrenheit 451" to be mindless drones who are easily and cheaply amused. Clowns are associated with little children and simple amusements. Clowns are not associated with sophisticated humor. Bradbury is showing the reader that these people in the book's society are unsophisticated and have the intellectual level of small children. On a slightly deeper level of interpretation, the White Clown, represents the leaders of Ray Bradbury's world. Not only the current world, but the past as well. The main leader from the past he's lumping into this one symbol is probably Hitler, who led a society that burned books. In the U.S., Bradbury is probably referring mostly to Sen. J. McCarthy who led the Senate committee investigation allegations of Communism in the U.S. and in so doing, unfairly persecuted many people. People in the arts were one of McCarthy's favorite targets and his persecution and witch-hunt style led many people to become fearful of saying or doing anything controversial, essentially censoring themselves. Bradbury, and many others, saw this as a very bad development and this book was one attempt to illustrate how wrong McCarthy was. The White Clown is how Bradbury saw these leaders - as buffoons.
THe author uses this program "White Clown" to portray TV as a tool of distraction for the populace as part of the totalitarian regime.
PS read the book 30+ years ago and still recall the line "don't you like White Clown" when I see the vast portions of today's programming.