If you go out of your way to get the gold sword before fighting Mothula in A Link to the Past, you will not be able to hurt him with your sword.
(It is possible to do this if you do Thieves' Town early for Titan's Mitts, beat Ice Palace using Bombos instead of the Fire Rod, and also beat Misery Mire. The giant bomb needed to see Venus only requires the crystals from IP and MM, so you can actually skip Skull Woods until you need the Fire Rod for Turtle Rock.)
The reason the gold sword cannot hurt Mothula is because it does the same damage type as spike traps (class 4 damage). Mothula is immune to class 4 damage because otherwise it would be killed by the traps in its room.
Incidentally, Mothula is the only boss that can be damaged by bees.