I've yet to come across something like this, though here are some thoughts on how such might be done...

If you use forums that make use of html frames, others could use frames as well to display the forum frame...a bit clunky though and I am unaware of forums that use frames out of the box.

You might be able to give out your forums database information ( perhaps with a read only mysql or sql account ) and configure forums on other sites to connect to your mysql/sql server to pull content from. I have not looked into such, but (beside security concerns) that seems like it might be the best route to look into.

Basically, they would have the forum software pointing to your database for data purposes...giving them the ability to display your forums on their site.

Are you trying to do something like these guys do? Check their remotely hosted email form. Bravenet - Web Hosting, Free Web Hosting and Web Tools

Email Forms Service by Bravenet
Yes, this is exactly what I'm looking to do...

Any suggestions?

Broadly, it's almost identical to what Bravenet is doing. You sign up for a membership, and you can have a remotely hosted form on your site just by copying and pasting a couple lines of code. Members would be able to specify what fields/etc. they want in the forms.