I want to make the Activity tab only visible to Friends, so I set the "permissions" value of that tab in the profileviewtabs.config file to "Friend".

This prevents non-friends from seeing that Activity tab, but the first visible tab doesn't open. Also, non-friends are also not able to click on the tab buttons to open a different tab.

Is there a way to have a tab open when a non-friend views a profile without having a tab with no permissions on it be first in the profileviewtabs.config file?

The way it works is: Any permission that is empty, will copy the permission of the one above it. But if the first one is empty, it defaults to an All Users type permission.

There are a few ways you can get around this. You can make a different tab be the first one in the list, and give it the empty permission. Or you can give all the other tabs the Auth permission, which allows all registered users to view it.

Having a tab visible, but not be able to open makes me think you made a syntax error somewhere in the profileviewtabs.config file or a javascript error on the page.