It would kill the forums. Plain and simple.
People just stops browsing and making threads they just ask everything there, and those logs get eventually wiped, everything becomes ephemeral which is everything the forum is not meant to be.
I've seen it before. it's much harder to moderate cos you have to be there 24/7, and it's also very heavy on server resources when you have many users, tons of ajax calls hitting the database can't be cached if it's real time. I know this has likes but I wouldn't do it.
I know some people don't like it but a
https://retrogametalk.com/discord is the only thing I can offer for chatting purposes.
/sorry