You'll need something like this in your root .htaccess file of your primary site (the one where the content used to be):
RewriteRule ^forum/(.*)$ http://old.boguswebsite.com/forum/$1 [R=301,L]
That will do a permanent redirect of all URLs in the forum directory to the same URL on your archive subdomain.
That solution assumes the all the old URLs were in a single directory. If they weren't, it gets a bit more complicated, but it's still doable.
And also of note is that mod_rewrite calls in .htaccess files only work a certain number of directories deep. So if your server is set to 3 directories deep and the URL is 4 directories deep, then your mod_rewrite rule won't execute. [There are workarounds for that too, if it's an issue.]
OR (slightly better)… Create an empty directory ("/forum/") where the URLs used to come from and put this in the .htaccess in that directory…
RewriteRule ^(.*)$ http://old.boguswebsite.com/forum/$1 [R=301,L]
That has two advantages: 1) it only executes for URLs in that directory, and 2) it gives you one more level of directory depth.