Do iPhone games which replicate the exact functionality and rules of popular proprietary board games need special permission (copyright, trademark, etc) from the relevant companies (Mattel, Hasbro, Parker Brother, etc) before being submitted into the App Store for approval?
Yes. Both the board game copyright holders and Apple will be pretty strict about the copyright for the content you are putting in your App.
There are examples of open source projects that replicate board game functionality (but not trademarks etc), but those projects are not designed to make money (as I assume your iPhone app would be).
Like many things involving intellectual property and the legal system, there isn't an easy answer to this.
Strictly speaking, you can't copyright game mechanics. However, it can be very difficult to 'clone' a lot of games in a recognizable fashion without infringing on trademark or copyright in the look or presentation of the game. This is setting aside any other ethical considerations of cloning existing games.
Unless it's really really blatant, Apple would probably approve it. However, if it's recognizable enough, it's probably going to get a cease and desist from the IP owner of the game you are cloning.
So, making a game with the mechanics of Monopoly that's recognizable as a Monopoly clone is asking for a smackdown. Making a game of some obscure boardgame from a long out of business company is "safer" in that it's rather unlikely somebody is going to spend the money to sue you with a dubious chance of winning.