Majora's Mask has without a doubt some of my favorite dialogue in the series. While I can hardly remember any lines from other Zelda games, like TP and SS, some of MM's lines will stay with me forever. Choosing one will prove to be difficult.
There's always the simple yet powerful dialogue from the Moon Children, such as “Your friends… What kind of… people are they? I wonder… Do those people… think of you… as a friend?" and "What makes you happy... Does it make… others happy, too?".
The Happy Mask Salesman has some great lines, too, such as the legendary "You've met with a terrible fate, haven't you?" and the aphorismic "Whenever there is a meeting, a parting is sure to follow. However, that parting need not last forever... Whether a parting be forever or merely for a short time... That is up to you."
I also quite like this quote from Igos du Ikana: "Haven't you begun to understand? The kingdom being ruined and us left in this state... Isn't it petty, little battles like this that have caused it?" This line is the moment where Igos transforms into a truly admirable character, having found realization as to why his kingdom is in the state it is. Knowing he can't remedy it himself, he gives Link the Elegy of Emptiness so that he may be able to stand a chance.
But even after all of these, I think my true favorite quote is as follows: "To die without leaving a corpse... That is the way of us Garo.". Subsequently the Garo that speaks that blows himself up. I can't even begin to describe this. This single piece of dialogue and the action that occurs immediately after solidifies the Garo as the most badass race in the Zelda series (imo).