Is it possible to remove a friend relationship between two FB users through the API? I'm thinking that it's not, but (if not) is it possible to at least bring up a dialog that would let the user request the unfriending, similar to how the Friends Dialog (http://developers.facebook.com/docs/reference/dialogs/friends/) lets a user send a friend invitation?
It is not possible through the API. Facebook is like the mafia - you can get in. but there's no way out.
Simialar to this question: Any way to unfriend or delete a friend using Facebook's PHP SDK or API?
Also, it is against the terms of service for facebook apps to ask people to unfriend. There was a BurgerKing prootional app that famously ran afoul of that after going viral. http://www.insidefacebook.com/2009/01/14/whopper-sacrifice-shut-down-by-facebook/
Let friends unfriend on their own time.