The phrase to say prunes means: to speak the plural noun prunes aloud in order to form the lips into an attractive shape.

This phrase is particularly associated with portrait photography—as in the following from the Ottawa Herald (Ottawa, Kansas, USA) of Tuesday 5^th January 1965:

Prunes Have Place In Good Menus

Say “prunes,” with a long “ooooo.”
Some photographers recommend “prunes” as the word that will bring pleasing facial expressions when the shutter clicks.
The U. S. Department of Agriculture also recommends that you say “prunes” when you order your groceries. Having prunes for breakfast, or any other meal, will do the same thing as saying the word—it’ll bring pleasing facial expressions. They’re so good.

The following explanations are from Why We Say “Cheese”: Producing the Smile in Snapshot Photography, by Christina Kotchemidova, Instructor in Communication Studies at New York University, published in Critical Studies in Media Communication (Taylor & Francis Group) of March 2005:

The “Say cheese” smile characteristic of contemporary popular photo culture has a history. Although its exact starting date cannot be established, we can easily contrast it to the invariably serious facial expression of nineteenth-century photos. Victorian families wishing to appear happy simply posed in front of the family property. The typical expression in early group photos and cartes-de-visite followed traditional European portraiture, which depicted solemn faces, occasionally softened by a Mona Lisa curve of the lips. In the fine arts a grin was only characteristic of peasants, drunkards, children, and halfwits, suggesting low class or some other deficiency. The only toothy smile in American nineteenth-century illustration art belongs to Huckleberry Finn—an Irish peasant child. Etiquette codes of the past demanded that the mouth be carefully controlled; beauty standards likewise called for a small mouth. Accordingly, the first photo studio in London, established in 1841, adopted the locution “Say prunes” to help sitters form a small mouth.

The text containing the earliest occurrence of the phrase to say prunes that I have found supports the explanations given by Christina Kotchemidova; this text is The Pencil of Nature, by Andrew Winter, published in The People’s Journal (London, England) of December 1846—the author gave advice on how to look best when photographed by the Daguerreotype process:

We wish ladies would be a little less prim on such occasions. It is quite melancholy to see the care they take to brush their hair, and apply that abomination, fixiture, to make it “look nice;” whereas, if a good breeze had broken it up into a hundred waves, the effect in the Daguerreotype would have been infinitely more beautiful. And let them by all means abjure the system of making up a face for the occasion. The effect is painfully transparent. The mouth, so expressive in all faces, in these portraits is nearly always alike; and for the simple reason, that we put its muscles into attitudes which are not at all natural to it—we substitute a voluntary for an involuntary action; and, of course, stiffness is the result. If the ladies, however, must study for a bit of effect, we will give them a recipe for a pretty expression of the mouth—let them place it as if they were going to say prunes.

The English novelist Charles Dickens (1812-1870) elaborated on the phrase in Little Dorrit (London: Bradbury & Evans, 1857)—cf. ‘prunes and prism(s)’: meanings and origin:

“Amy,” said Mr. Dorrit, “you have just now been the subject of some conversation between myself and Mrs. General. We agree that you scarcely seem at home here. Ha—how is this?”
A pause.
“I think, father, I require a little time.”
“Papa is a preferable mode of address,” observed Mrs. General. “Father is rather vulgar, my dear. The word Papa, besides, gives a pretty form to the lips. Papa, potatoes, poultry, prunes, and prism, are all very good words for the lips: especially prunes and prism. You will find it serviceable, in the formation of a demeanor, if you sometimes say to yourself in company—on entering a room, for instance—Papa, potatoes, poultry, prunes and prism, prunes and prism.”

The other early occurrences of the phrase to say prunes that I have found are as follows, in chronological order:

1-: From How a Woman Goes to a Fire, published in The Napa Daily Register (Napa, California, USA) of Tuesday 15^th July 1879:

She seizes a hand glass and takes a retrospective view of her head and all thereto appertaining. Now she adjusts her veil so that the lower edge will come exactly to the tip end of her nose; then she powders a little so that her face will not “shine like a bottle,” smooths down the corners of her mouth, says prunes, and puts on her avenue smile.

2-: From The Latin Lesson: Boy and Girl, a dialogue between Tommy and Sybil, published in Blackwood’s Edinburgh Magazine (Edinburgh, Midlothian, Scotland) of November 1881:

S. What a way to talk!
T. It don’t do for a girl. You have to say “prunes” and “precision” all day to make your mouth pretty.
S. Tommy, you are exceedingly silly; and it’s better to say “prunes” than to chew grass.

3-: From an article giving advice on being photographed, published in The Cincinnati Enquirer (Cincinnati, Ohio, USA) of Sunday 12^th August 1883:

If your mouth is too large say “prunes,” and let your lips remain in the position in which they are left when the last letter is sounded. If the reverse is the case, and an enlargement is desired, say “cabbage,” and heavens knows it will be large enough. An intellectually superior look can be effected by saying “besom.” Poesy may be expressed by using the word “soften.” “Coward” will give firmness.

4-: From Interesting Gossip of the Day, published in The Sun (New York City, New York, USA) of Sunday 9^th December 1888:

Said an old observer: “Marguerite Fish stands on an eminence as being about the only woman on the stage who persistently ‘mugs’ through a part until nobody has any idea at the end of the play what her real face looks like. Most women simply ‘mug’ to the extent of enhancing their beauty. If they find that their mouths look pretty when they say ‘prunes’ they murmur ‘prunes’ for the rest of their natural lives. They ‘mug’ in that direction only. Miss Fish, however, twists her face into all sorts of queer contortions, purses up her lips, scowls and makes more grimaces in a minute than any other actress I ever saw in a week. She seems to grow healthy on it.”

The phrase to say prunes is also associated with kissing—as in the following two texts:

1-: From In the Limelight, by Herbert Farjeon, published in the Sunday Pictorial (London, England) of Sunday 13^th October 1929:

Kissing-Time.—The first performance of “A Yankee at the Court of King Arthur” was received at Daly’s Theatre without remonstrance. There was much speechifying at the end, in the delirious course of which William Fox, an American comedian, kissed the English actor, Sam Livesey. This was preferable to the kissing that went on during the actual performance, with jokes to taste: such as teaching damsels to say “prunes” before they osculate. After all, we were not attending a twice-nightly revue in a dockside music-hall.

2-: From How you kiss and make up, published in the Daily Mirror (London, England) of Wednesday 25^th August 1971:

Say ‘Prunes’
FINDING it hard to say “Sorry” even after nearly thirty-six years of marriage, I resort to a trick we used to do in our courting days.
I just pucker up my lips and say “PRUNES” a few times. It works like magic. He puckers up too and we kiss and make up.—(Mrs.) H. H., London. N 16