In Act 1, all four of the Tyrones discuss are discussing "Papa's snoring." During this discussion, Jamie makes the first of a great many Shakespearean allusions in the play:
Jamie:
I heard him too.
He quotes, putting on a ham-actor manner.
"The Moor, I know his trumpet."
His mother and brother laugh.
Tyrone:
Scathingly.
If it takes my snoring to make you remember Shakespeare instead of the dope sheet on the ponies, I hope I'll keep on with it.
This is a reference to a fairly innocuous line from Othello, Shakespeare's tragedy about a Moorish military commander named Othello and the infidelity of his wife, Desdemona. (A "Moor" was a European name for Muslims from North Africa and other places around the Mediterranean during the Middle Ages; the term was never used by the people themselves.) In Act 2, Scene 1, Iago, Othello's malicious aide, goes to Cyprus in search of Othello. He hears Othello beginning a general celebration in the city for his marriage, complete with music, and calls out: "The Moor! I know that trumpet!"
Jamie's quickness with this reference and Tyrone's response clarify their characters. Jamie wants, at once, to impress his father and make fun of him. Tyrone, a washed-up actor, wants his children to learn Shakespeare, as a marker of their class, distinction, and education. So, Jamie quotes Shakespeare to chide his father's snoring: he knows his father's trumpeting snores like Iago knows Othello's trumpet. This reference, which is not one of the more well-known lines from the play, shows Jamie's quick wit and extensive knowledge of Shakespeare (as well as Edmund's and Mary's, since they immediately get the reference).
O'Neill's stage directions at the beginning of the play describe each member of the family in great detail. Jamie's description, particularly the description of his face, makes reference to German folklore:
[Jamie's] nose is unlike that of any other member of the family, pronouncedly aquiline. Combined with his habitual expression of cynicism it gives his countenance a Mephistophelian air. But on the rare occasions when he smiles without sneering, his personality possesses the remnant of a humorous, romantic, irresponsible Irish charm––that of the beguiling ne'er-do-well, with a strain of the sentimentally poetic, attractive to women and popular with men.
Jamie's "countenance"—his face—has a "Mephistophelian air." This refers to Mephistopheles, also known as Mephisto, who is a demon common in many folktales from Germany and other cultures. He is most prominently featured in the myth of Faust. In that story, Dr. Faust, a highly successful but unsatisfied academic, makes a deal with the devil (via Mephistopheles) in which he exchanges his soul for wealth, knowledge, and pleasures. The most famous version of this story is in the play Faust by Johann Wolfgang von Goethe, often considered the finest classic in German literature.
Mephistopheles is humanoid but has stereotypically devilish features, particularly his long, pointed face. He is also charming and manipulative, able to convince people to sell their soul in corrupt bargains. This is a rather ungenerous characterization of Jamie.
In Act 2, Scene 2, Edmund asks his father whether he prayed for Mary. His father says he did, and Edmund, based on this, comes to a conclusion:
Edmund:
Then Nietzsche must be right.
He quotes from Thus Spake Zarathustra.
"God is dead: of His pity for man hath God died."
Edmund references the German philosopher and theorist Friedrich Nietzsche. Specifically, Edmund quotes his book Thus Spake Zarathustra, a book of stories and tales about Zarathustra, also known as Zoroaster, who is a deity who was revered in multiple cultures in the ancient Middle East. The book describes a wide variety of subjects but, most famously, contains Nietzsche's arguments about the death of God. Nietzsche claimed that, after the Enlightenment, a belief in God had become "unbelievable," as a scientific, secular worldview became more widespread. Thus, because God exists only if people believe in him, Nietzsche claimed "Gott ist töt": God is dead. Philosophers before Nietzsche, especially in Germany, had discussed the death of God for decades, but Nietzsche described the claim most extensively and memorably.
Edmund alludes to this claim from Nietzsche sardonically. He asks whether his father prays for his mother, but since Mary is still wracked with addiction and anxiety, Edmund concludes that God must be dead. This is a much more personal interpretation of Nietzsche's claim, which primarily concerns belief in God on a societal level, not an individual level. Still, Edmund quickly and smartly references Nietzsche.
The next stage direction is curt: "Tyrone: Ignores this." Tyrone wants his children to memorize Shakespeare, not Nietzsche. Just after this, Tyrone admonishes Nietzsche as "rotten to the core." Edmund's ability to quickly and accurately quote Nietzsche is an intentional bit of rebellion against his father.
Tyrone, in Act 2, Scene 2, begrudgingly agrees to give Edmund money for carfare to the doctor. Tyrone tries to get Edmund to thank him, using a quotation:
Tyrone:
[...] Thank you.
He quotes.
"How sharper than a serpent's tooth it is––"
Edmund:
"To have a thankless child." I know. Give me a chance, Papa. I'm knocked speechless. This isn't a dollar. It's a ten spot.
The reference is to King Lear, Shakespeare's tragedy of the legendary king of England and the conflicts among his descendants. Tyrone and Edmund quote from Act 1, Scene 4, when the king is referencing the unfaithfulness of his daughter Cordelia. Tyrone quotes it to try and remind is son that he ought to thank him, as Cordelia ought to have been more thankful to Lear.
This allusion is significant, for one, as a continuation of the father and son's practice of quoting Shakespeare to each other. Edmund is notably more willing, here, to acquiesce to referencing Shakespeare, after being more rebellious by referencing Nietzsche earlier in the scene. Tyrone's quotation has a rote tone, as if he has often admonished his children with this line from Lear when they have been thankless. Note as well that Edmund is the name of the main antagonist of King Lear, whose attempts to play Lear's children off each other and to usurp his own father, the Duke of Gloucester, form the main conflict of the play. Tyrone may have referenced King Lear with this in mind, meaning it as a subtle jab at his son. Taken together, this allusion advances the emotional relations of Tyrone and Edmund in the play, deepening their simmering tension.
In Act 4, Tyrone and Edmund have an argument over whether Edmund should be drinking. Tyrone is trying to tell Edmund that he might as well go to sleep because he is sick and his body needs rest. He explains this by making an allusion to Shakespeare:
Tyrone:
Why can't you remember your Shakespeare and forget the third-raters. You'll find what you're trying to say in him––as you'll find everything else worth saying.
He quotes, using his fine voice.
"We are such stuff as dreams are made on, and our little life is rounded with a sleep."
Edmund:
Ironically.
Fine! That's beautiful. But I wasn't trying to say that. We are such stuff as manure is made on, so let's get drunk and forget it. That's more my idea.
Tyrone quotes from The Tempest, one of Shakespeare's latest plays and one of his most magical. The play concerns a wizard and the former Duke of Milan. The Duke, named Prospero, has been ousted by his treacherous brother and has fled to a deserted island with his daughter, Minerva. Tyrone references a moment in Act 4 of The Tempest, just after a play-within-a-play. Prospero is speaking to Ferdinand, a survivor of a shipwreck on the island who is to be married to Miranda. Ferdinand acted and danced in that play, and Prospero is advising him to have a full night of sleep after putting on a full performance that day.
This, one of the last Shakespeare allusions in the play, is a gentler and more whimsical one than many of those before. Tyrone, now drunk and more emotionally soft, leans into a poetic turn of phrase about a peaceful sleep from a peaceful scene in Shakespeare's play. Using the different tone of the allusion, O'Neill emphasizes how alcohol dulls Tyrone's anger and aggression.
Edmund, though, responds by humorously interpolating the allusion: we are not made of dreams but of manure. This is meant to express the same sentiment as the idiom "ashes to ashes, dust to dust," but in a coarser way: we are made of what was once waste and what will soon be waste again. Edmund understands that his father is telling him to stop drinking and to get some sleep. But Edmund, who is aware of his tuberculosis and his mortality, is not especially concerned with preserving his body. Very aware of the fact that his body may soon be "such stuff as manure," he tells his father to get drunk anyway.