It's not clear that a forum administrator would need to do anything under the GDPR except respond to emails with the standard template: "The forum does not collect or process any personal data."
Technically, I suspect, this would be true. The GDPR and the right to be forgotten are subtle on this. If a user chooses, unprompted, to share PII it's not clear that collection has taken place. Imagine a user, out of the blue, uploads her bank account information to a forum and others take the credentials and steal or her money. Would anybody seriously believe that the business should be liable for failing to secure the PII data? The other question is whether such data can be said to be processed. Clearly the forum is not processing the PII data as PII data. It's likely the case that the business doesn't know that any given forum contains PII data.
The GDPR also gives businesses a lot of lee way here. Erasure requests can be rejected under "freedom of information", if they cause undue burden a fee can be demanded, or if they're just frivolous they can be outright refused [1].
Admittedly this is speculation. European regulation is rules-based and a lot of leeway is given to regulators. The right to be forgotten is probably the least concrete aspect of the law. It's not clear how it intersects with user-generated content because it's not clear that PII is even being collected here. In the end, I suspect the regulators and the courts would probably be open to good faith efforts towards compliance. This might take the form of removing account data (username, emails, profile pictures) but leaving the forum posts up. If a specific forum post is believed to contain PII the business might ask the user to explain how the data in the post could be used to personally identify them as a data subject.
[1] https://ico.org.uk/for-organisations/guide-to-the-general-da...