Is there a way to upgrade from 1.1.5 to 2.x beta without affecting the current forum?
No, the upgrade between SMF 1.1.5 and SMF 2 Beta is a large one, it basically overrites all of your forums files, meaning that all mods will be uninstalled. Also many of the mods for SMF 1.1.5 are not available for SMF 2 Beta, so you may not be able to keep on using some of the mods that you have. This is the same for custom themes as well.
Having said that, all of your forum data - topics,message,posts and etc, will remain intact.
If you do want to upgrade, then I recommend that you have a read of the documentation:
Upgrade SMF (http://docs.simplemachines.org/index.php?board=3.0;sort=subject)
I would also recommend that you perform the upgrade for a local/development version of your website, before you make it like. You should also backup everything.
K thanks.