There could be country-specific laws which might entail such a requirement, but in general, there is no such obligation on moderators. By way of analogy, certain information is classified top secret and only people with a certain level of security clearance from the US government are allowed to see such information. Suppose some rogue agent posts that kind of information on a website, and a diligent moderator recognizes that this material is illegally posted and should be deleted immediately. There is no requirement that moderators must hold a high-level security clearance on the remote chance that a rogue agent will post illegal material and the moderator might see stuff that they are not supposed to see. The same would hold for porn, blasphemy, and politically disallowed statements.
In general, it is not a crime for a person to have seen illegal material, thus it is not a crime for a child to have seen a filthy picture (although there could be some repressive regime where it is illegal for anyone to see such material). Now, it is not entirely inconceivable that some regime has vague and hyper-protective laws that prohibit minors from "holding a position of trust and authority", but certainly there are no such laws in the US, Canada or most of Europe that I know of. There can be real restrictions on minorsw, such as the (non-)ability to form contracts, but the job of being a moderator doesn't involve forming contracts.