Mana burn is no longer a part of the game as the other answers point out so well. However, I feel like I have a definitive answer to "Why was the rule added?"
In episode #4 of his game design podcast - Games With Garfield - Richard himself and Skaff Elias comment on the creation of mana burn.
transcript from 05:27 to 06:25
SKAFF: Mana burn really wasn't even in there in the beginning, for the first couple of years we played the game and it kind of crept in because we didn't want people to just go... We needed a mana pool to make sense out of casting... exactly how a card was cast, to try and formalize that, and then some people who were just trying to be jerks (maybe I was one of them) would just dump all their mana into their mana pool so that, you know, it couldn't be disrupted, cause certain things disrupted it back at the time. And, anyway, to sort of prevent that and to prevent people to ever wanting to do that, mana burn was introduced into the game. The clearing of mana out of the pools, and maybe that wasn't good enough so maybe actually take some damage from it. And then from there it took on a life of its own and actually became part of the play of the game.
And Richard Garfield adds:
transcript from 6:26 to 6:47
RICHARD: Certainly, back in the old days there was a point where you could technically, at the start of you turn, just tap all your land and then you'd have all your mana for the whole turn. And the only difference would be that your opponent couldn't tell how much it was, was a pain to keep track of and so forth and so... Mana burn eventually was an answer to that (...)
So, to sum up, it seems that mana burn was created as a game design mechanical fix to avoid memory issues and to avoid some weird situations in the original spell timing rules.