I recall reading in the changelog that updates have been made to templating... Is there anything in the 1.6.5 update that will break my current theme or templates?
in general, No. if you are upgrading from 1.6.4 by using changed files package then your custom
templates do not get disturbed. if you are upgrading from an earlier version by using the full 1.6.5
package AND if you have custom templates for myBB's default theme then they might break ...
They don't really "break" - they just have to be updated. If you follow the upgrade instructions on the wiki you'll be fine.