The cause of death was Parkinson's disease, according to publicist Richard Hoffman.
Jones made his film debut in 1956's "Somebody Up There Likes Me," which starred Paul Newman as boxer Rocky Graziano. The next year he appeared in the Elvis Presley movie "Jailhouse Rock."
He also starred on Broadway in such works as "There Was a Little Girl," with Jane Fonda, and the hit "Under the Yum-Yum Tree," also starring in the 1963 Jack Lemmon film.
Jones originated the performance of Bobby in the 1970 production of Stephen Sondheim's "Company."
But he had his greatest success as a Disney player, usually playing genial fathers or bemused leading men.
His Disney debut was in 1965's "That Darn Cat!", in which he played an FBI agent who solves a kidnapping with the help of the title character (and Hayley Mills, another Disney regular). He followed it up with such films as "Monkeys, Go Home!" (1967), "The Horse in the Gray Flannel Suit" (1968) and the extremely popular "The Love Bug" (1969), about a Volkswagen with a mind of its own.
The latter spawned four sequels.
After a number of other film and TV roles, Jones returned to the stage in the 1980s, notably starring in the one-man show "St. John in Exile," about the apostle.
Jones was born in Decatur, Alabama, in 1931. He served in the Navy during the Korean War. Later in life, the actor -- who had become a devout Christian after a drunken driving accident
almost killed him -- founded Christian Rescue Committee, now known as Christian Rescue Fund. The organization works to "rescue persecuted Christians and Jews," according to its website
Jones is survived by his wife of 42 years, Lory Basham Jones, three children, eight grandchildren and three great-grandchildren.