Q. If names are listed last name first and first name last, after which name does the suffix (Jr., Sr., III, etc.) follow? How
is this formatted?
A. Commas, even when not used with the noninverted form, are always used with inverted names, which should appear in the following
order:
Surname, Given Name, III
Deer, Jim G., Sr.
[This answer relies on the 17th edition of CMOS (2017) unless otherwise noted.]
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 6.43 of the seventeenth 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).
[This answer relies on the 17th edition of CMOS (2017) unless otherwise noted.]