Understanding the events of a play and your character's role in these events will greatly help you to memorize lengthy dialogues, and any lines you may have. This technique of reading and comprehending the "whole" play to memorize the "parts" is accurately called whole-part memorization.
Whole-part memorization might be the best way to memorize large amounts of lines, according to the Association for Psychological Science. They refer to an actor's technique of memorization as "active experiencing," or in other words, not just knowing the lines, but the meaning behind them. Memory is strongest when associated with emotion and experience, and a great actor uses techniques that help him "feel" the lines.
Imagine the play as an intricate story. Know the script through and through, even beyond what your part requires.
Immerse yourself in the motivations and personalities of other characters--you will better understand the dialogue between them.
Take notice of what occurs immediately before and after your scene (including plot points and dialogue). This will aid in your memorization of what occurs during your part.