The phrases Queen Elizabeth is dead and Queen Anne is dead are used humorously and ironically to denote old news—usually with the implication that the person whom the speaker is referring to is simply stating the obvious or restating a well-worn or accepted truth.

—Cf. also origin of ‘old chestnut’ and of French ‘marronnier’.

The phrases Queen Elizabeth is dead and Queen Anne is dead refer, respectively:
– to Elizabeth (1533-1603), Queen of England and Ireland, who reigned from 1558 to 1603;
– to Anne (1665-1714), Queen of Great Britain and Ireland, who reigned from 1702 to 1714.

The phrase Queen Elizabeth is dead occurs, for example, in The Old Journalism, by Russell Baker, published in The New York Times (New York City, New York, USA) of Saturday 3^rd July 1976:

On Aril 19, 1775, the shot heard round the world was fired at Lexington. It was July 3, 1776, when King George heard it in London.
“That sounds like a very old shot,” he said to his equerry.
“From the sound of it,” said the equerry, “I’d say it was fired about one year, two months and two weeks ago.”
[…]
The King fumed. It was impossible to govern sensibly with such haphazard communications.
[…]
Then was the King sore wroth until his doctor arrived with the latest news. “You mean that no King since Henry VII can possibly be sore wroth?” he demanded.
“Quite impossible,” said the doctor. “The affliction was wiped out by linguistic revision during the reign of the late, great Queen Elizabeth.”
The King wept. Tremendous events were occurring all over his vast realm, but poor communications kept him ignorant of them for years.
“Why do you weep, Your Majesty?” asked the equerry.
“Haven’t you heard?” asked the King. “Queen Elizabeth is dead.”
The equerry had, in fact, heard the news several years before, but thought it indiscreet to show that his information was more current than the King’s.

The phrase Queen Anne is dead occurs, for example, in Prenno on Telly, by David Prentice, published in the Liverpool Echo (Liverpool, Merseyside, England) of Saturday 15^th January 2005:

At least Hansen’s views come across as authoritative and insightful—regardless of pronunciation.
Unlike his former central defensive partner.
“McFadden’s gone past the three French players there,” said Lawrenson, who can also tell you that Queen Anne is dead, night follows day and bears defecate in the woods.

—Cf. also the phrases Queen Anne’s fan and Queen Anne front and Mary Ann back.

The phrase Queen Elizabeth is dead is first recorded in A Complete Collection of Genteel and Ingenious Conversation, According to the Most Polite Mode and Method Now used At Court, and in the best Companies of England (London: Printed for B. Motte, and C. Bathurst, 1738), by the Irish satirist, poet and Anglican cleric Jonathan Swift (1667-1745):

Lady Smart. And pray, What News Mr. Neverout?
Neverout. Why, Madam, Queen Elizabeth’s dead.
Lady Smart. Well, Mr. Neverout, I see you are no Changeling.

Note: This book, published in 1738, but written in the first decade of the 18^th century, is a satire on the use of clichés: its purported author, Simon Wagstaff, declares “that there is not one single witty phrase in this whole collection which hath not received the stamp and approbation of at least one hundred years”.

The following story about the city of Coventry, Warwickshire, England, published in Yorick’s Jests: Being a New Collection of Jokes, Witticisms, Bons Mots and Anecdotes (Dublin: Printed for J. Williams, 1774), mentions the fact that Queen Anne is dead:

The council-chamber of the same corporation having occasion to be beautified, the Mayor for the time being, which was soon after the accession of George I, undertook to superintend the workmen employed in that business. As is usual, an inscription was put up, after the whole was completed, mentioning the year, &c. in which the room was ornamented; but the wise Mayor perceiving the words Anno Domini, immediately sent for and abused the painter for committing such a gross blunder, as putting Anno Domini; when, says he, don’t you know that Queen Anne is dead, and therefore it should be Georgio Domini.

The earliest occurrences of the phrase Queen Anne is dead that I have found are as follows, in chronological order:

1-: From The Heir at Law (Dublin: Printed by T. Burnside and George Folingsby, 1798), by the English playwright and theatre manager George Colman (1762-1836):

Lady D. But why so loud?—I declare the servants will hear.
Lord D. Hear! and what will they hear but what they know? our story a secret, Lord help you!—tell ’em Queen Anne’s dead, my Lady. Don’t every body know that old Lord Duberly was supposed to die without any heir to his estate—as the Doctors say, of an implication of disorders; and, that his son, Henry Moreland, was lost, some time ago, in the salt sea.

2-: From Nouveau Dictionnaire François-Anglois & Anglois-François (London: Cadell & Davies; F. & C. Rivington; Longman, Hurst, Rees, & Orme – 1805), by Louis Chambaud and J. Th. H. Des Carrières:

Ce [tient quelquefois lieu du pronom cela] It, that. C’est fait, It is done. C’est fort bien fait, That is very well done. […] † C’est renouvelé des Grecs, † Queen Anne is dead.
[…]
Deviner [juger par voie de conjecture] To guess at, to conjecture, to hit upon, to prejudge, to smoke, to find out. Vous avéz [sic] deviné ma pensée, You have hit upon my thought.
† Il devine les Fêtes quand elles sont venues (il parle d’une chose connue de tout le monde, comme d’une nouvelle découverte), He tells by way of news that Queen Anne is dead.

Notes:
– In this dictionary, the dagger sign (†) indicates “a Proverb, or proverbial form of speech”.
– The French phrase C’est renouvelé des Grecs (literally: It is renewed from the Greeks) is glossed as The old thing over again! in A New Dictionary of Quotations from the Greek, Latin, and Modern Languages. Translated into English (London: John Farquhar Shaw, 1858).
– Il devine les Fêtes quand elles sont venues (il parle d’une chose connue de tout le monde, comme d’une nouvelle découverte) translates literally as: He foresees the celebrations when they have happened (he talks of something known to everyone, as of a new discovery).

3-: From Neglect of the Ladies, a letter to ‘Gossip’, by ‘Clementina’, published in The Gossip; A Series of Original Essays and Letters, Literary, Historical, and Critical; Descriptive Sketches, Anecdotes, and Original Poetry (Kentish Town [London]: J. Bennett) of Saturday 16^th June 1821:

“But look!” said Seraphina, tearing her glove as she hastily stripped it from her hand to take up the number, “here is an article on Gloves!” Well, said I, gloves are worn by belles as well as beaux, they can’t exclude us from that article. […] Nothing relating to our sex, except something at the end about a statute passed in the first year of the reign of Cupid, which entitles a lady to a pair of gloves for kissing a sleeping gentleman. How technically soporific! And then to tell us that gloves rhyme admirably to loves and doves! When my femme de chambre read this she exclaimed, “Queen Anne’s dead!”