Do you add a comma before Jr or Sr or III in a name?

No comma is necessary before "Jr.", "Sr.", and the like. No comma has ever been necessary before "III", "IV", etc. From the Chicago Manual of Style (http://ow.ly/gcv0):

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).