What happens if I update an app using its built-in ability that I built/installed through MacPorts? Will I break MacPorts ability to manage and update that same app?  Would it even allow me to do so?

For example, both MacVim and iTerm2 have a "Check for Updates..." function built into them.  If I built/installed those apps through MacPorts, and then updated them using the "Check for Updates", would I lose the ability for MacPorts to manage the app's version?
Probably not as Macports keeps information about what versions are installed etc in a database /opt/local/var/macports/registry/registry.db So when a new version comes out from Macports it will compare with this db.

However if you have done an update otherwise then why do you want macports to update it as the versions won't be as expected.