There is no hard and fast rule here. I think you need to handle these things on a case by case basis.
If the comments are closely related to the original question and you have the capability then by all means edit your answer to cover the new feedback.
If they seem unrelated or beyond the scope of the question answered (or your ability to answer) then you should certainly suggest that they open a new question to cover that topic or point them toward an existing answer that covers their concerns.
I would hesitate to involve the mods on anything similar to the aforementioned. There's no point in adding to their burden with something you can easily handle yourself or with the help of the community
If the comment could be construed as rude or offensive, then yes by all means involve the moderators, if not you can certainly take one or more of the actions mentioned above to help the member reach a solution.