Jr., Sr., III

Q. John Smith Jr. or John Smith, Jr.? John Smith III or John Smith, III?

A. Traditionally, it would be John Smith, Jr., and John Smith III. But beginning with the fourteenth edition of The Chicago Manual of Style (1993), the recommendation is to use no commas in either case (see paragraph 6.47 of the sixteenth edition):

John Smith Jr.

But please note that within text, if you decide to use the more traditional comma before Jr. or Sr., the function of the comma is to set off these abbreviations, so an additional comma is needed after the abbreviation if the sentence continues (as in my first sentence above).