I can move threads from one forum to another. Is there an easy way to move a whole forum from one category to another?
Using a new install of 1.6.5 with a dark theme by ShadowKage.
Yes - go into the ACP > Forums & Posts > ForumName > go into the dropdown and choose "Edit forum". Change the "Parent Forum" to the new category you want it to be in.
I hope this is what you mean -- and hope that this helps you out.
Edit : not seen above response 
you can change forum's parent forum in forum management (edit forum)
admin panel --> forums & posts --> forum management --> edit forum -->
Parent Forum <-- select other category
