Seriously, I can't imagine why you'd want to do this. The only valid reason I can think of would be if you're forced to because your web host supports php but not mysql, which would be odd, but not unheard of.
But even if that is the case, there are sites out there that let you set up a free forum running phpBB or something similar on their servers, so I don't think there's much need for a non-DB forum.
A forum is really a bit too complex to be trying to run from a flat file. But if you're determined to try it, I looked up on www.HotScripts.com,
and found a couple for you:
You'll probably find more if you look around, but those were the first two that came up.
I hope that helps.