Mana burn is an obsolete game concept in which a player would experience loss of life points for having excess mana in his or her mana pool when a phase ends. For instance, if a player tapped a Swamp to cast Dark Ritual, which would give that player in his or her mana pool, then cast Erg Raiders (which costs ), and then let the phase end in which Erg Raiders was cast but did not spend the remaining , that player would lose 1 life from mana burn.
Mana burn was almost removed in the Sixth Edition rules update, but Mark Rosewater fought to keep it in. Ten years later, he fought to have it removed. His reasons to remove it were that it would free up design space, do away with a rule that's confusing for new players and that it's a rule that wasn't pulling its weight.
He tried it out in design by having all designers play without mana burn for a month. It hadn't come up in playtests at all during the whole month. 
 In-world meaning
In storylines mana burn happens when a wizard holds mana and doesn't use it to cast the spell. Such a wizard can seriously suffer from it, be killed, or even be burned to ash if the amount of mana was too big. Some victims of mana burn can be also be turned into the Fallen.