An obvious answer is to use true <frame>s instead of an <iframe>.
The chat is in one <frame>. The forum board is in the other <frame>. When you change forum content, you do it by loading a different page into the forum's <frame> and the chat <frame> is undisturbed.
Old fashioned, maybe, but it works.
Failing that, you'll just have to use two separate windows.
An optimist sees the glass as half full.
A pessimist sees the glass as half empty.
A realist drinks it no matter how much there is.