You are not obliged or required to delete their posts, the only time that would come into play is if the posts contained any personally identifiable information; which, they shouldn't have and you can make it an explicit requirement in your T&C that users do not make any PII known. You can also add in your T&C that anything they post can be used continuously - how you word this is up to you as long as it's clear and understandable.
Individual items cannot generally identify a person, but several items together can. For example if a someone called John Smith joins your forum and uses their real name as their username, this alone cannot identify an individual because their may be millions of people called John Smith, but if you add in other information such as an email address, static IP address, DOB and so on - then you can identify that person as an individual.
The problem with the GDPR is that it makes it complexed when deciding how much of what kind of data can be enough to identify someone as an individual in the course of running a forum site. We then, as forum owners, become data controllers (and possibly data processors) so we then assume control of whatever data is supplied to us and we have to act accordingly in conjunction with GDPR.
The other problems this raises is keeping track of how that data is used, where it's stored, the security measures taken to protect it and then giving the individual rights to be forgotten, the right to have access to how their data has been used and the right of portability of that data. Not to mention cookies and 3rd party cookies and getting explicit permission to drop those cookies from an individual every time a cookie is dropped from wherever.
And so the show goes on - but you don't have to delete posts as that would cause disruption to conversations that other people are taking part in and may be relevant to the discussion. How you deal with the deletion of an account will depend on your own process for handling it.
It may be an opportune time to buy a wig because the prices of them will go up as demand increases from people pulling their hair out over this lol.
