Mana burn does not exist as a rule in MTG anymore. It was removed from the rules when they released the 2010 core set.
Back when the mana burn rule did exist, Time Stop still would cause a player to lose life due to mana burn. This is because mana burn happened automatically whenever a phase ended, and time stop caused the current phase to end. Mana burn did not use the stack, and was simply an automatic game action that occurred.