If I'm writing an article about John Smith III, I'll use his full name, including the suffix, when I first introduce him in the article. For example:
"The owner of the dog, John Smith III, was surprised at the size of the [...]"
After that, if I refer to him by surname, I don't use the suffix:
"When questioned about the damage, Smith was evasive."
I see and hear people include the suffix when referring to someone by surname, which seems wrong to me:
"Smith III continued on as if nothing was wrong."
Is there a formal rule that states to not use the suffix with surname references?