In the name John Smith III is III followed by a comma?

Not unless a comma is otherwise required. The commas around generational titles apply only to the abbreviations Jr. and Sr. when used following a full name e.g. "Henry Ford, Jr., was a major industrialist."

(This formal requirement has been increasingly omitted in colloquial text. The IRA no longer uses commas in text citations, and APA omits them except necessarily where the name is reversed.)