Using localStorage, the forum's content can be stored offline in the browser, which'd then get synchronized between online users with Peer.js.
There'd need a signalling server to introduce users to each other, but the server-side database load would be absolutely zero. And client-side localStorage has a limit of 10MB per domain, but I think that can be worked with if it doesn't grow excessively big, or maybe it can be circumvented somehow.
You could just go to your regular php file manager and upload an index.html, then go to yourhost.com/forum/ and the forum would be there, wholely javascript without page loads.