I played with some new folks recently, and they have a tradition of the dealer showing the burn card (the burn card is the card dealt between the hole cards and the flop, after the pre-flop betting). I have never seen this done before, and although it did nothing but help me (once the burn card was a three of clubs, and it told me that there was no way the guy across from me had a three in the hole, since I knew where all the threes were) but it’s for that very reason that every game I’ve ever played in has had the burn card kept face down.
That’s the only time a burn card is exposed, and even then it becomes a burn card after it’s been exposed. From what I know from working in a casino for six years, all burn cards are face down during the game.
But then there’s someting call “house rules”. If that’s a house rule, you either play by it or you don’t play at all. As long as you know about it beforehand.