Of American-English origin, the expression dumb bunny denotes a stupid person.

This expression seems to have originated in the slang of the flappers (i.e., the young women who, in the 1920s, showed freedom from conventions) and of their male counterparts.

The earliest occurrences of the expression dumb bunny that I have found are as follows, in chronological order:

1-: From the column The Conning Tower, by Franklin Pierce Adams (1881-1960), published in the New York Tribune (New York City, New York) of Wednesday 21^st September 1921:

As we interpret the lingo of the younger Scott Fitzgerald ¹ set, a dumb bunny is a goof with a part-time bean ².
They are back in town, that set, and Fifth Avenue looks like the Flappian Way.

¹ This refers to the U.S. novelist and short-story writer Francis Scott Key Fitzgerald (1896-1940), noted particularly for his depictions of the period between the end of World War I (1918) and the beginning of the Depression (1929).
² Here, goof denotes a guy, and bean denotes the intellect.

2-: From the column Home Brew, published in the Harrisburg Telegraph (Harrisburg, Pennsylvania) of Wednesday 12^th October 1921:

Speaking of intellect and the display of same, what could be more entertaining than the manner in which the motoring boys get their cars in the middle of the street at the first alarm of fire and keep them there until the last apparatus has gone by. We await with a deal of pleasure the day when a heavy fire truck will crush one of these dumb bunnies against the curb, where he belongs.

3-: From a special dispatch by Charles Brower, published in Every Evening (Wilmington, Delaware) of Tuesday 22^nd November 1921:

Washington, Nov. 22.—The world’s greatest society circus was held here yesterday. On the cards it was listed as “the second plenary session of the Conference on Limitation of Armaments.” The fate of France—and the world—was discussed there. […]
[…]
11 o’clock. Mr. Hughes is speaking. He is introducing M. Aristide Briand, Premier of France***** “O, look, there is Mrs. Harding, see, right there in the box! And there is Mrs. Coolidge—you poor dumb bunny, she’s the Vice-President’s wife.”

4-: From the column The Conning Tower, by Franklin Pierce Adams (1881-1960), published in The Boston Daily Globe (Boston, Massachusetts) of Wednesday 18^th January 1922:

FLAPPERS AND PHILOLOGISTS

Sir: The dumb bunnies of this town button the bottom buckle of their arctics, leaving the top three open, buffeted about their silk-stockinged calves, suggestive to me of the origin of the word flapper.
Freddie.
Portland, Me.

5-: From The English Language as She Is Spoke In New York by Younger Generation, published in The Lexington Herald (Lexington, Kentucky) of Thursday 9^th March 1922:

CELLAR-SMELLER—A young man who always turns up where liquor is to be had without cost.
CAKE-EATER—A small salaried male person who frequents teas and other entertainments and never makes any effort to repay his social obligations.
DARBS—A person with money who can be relied on to pay the check.
EGG—A hard-boiled cake-eater who lets a jane pay her own way into a dance hall. Egg-harbor—A dance hall where no admission is charged.
FLAT-WHEELER—A young man whose idea of entertaining a girl is to take her for a walk.
ERIE—Any person who lives in New Jersey—e.g., “A lot of Eries sitting around the hotel.”
OVER THERE—A person, male or female, who lives in Brooklyn—often used in warning where a taxicab is involved.
STRIKE-BREAKER—A girl put in to take the place of a young man’s regular girl when she is away.
FINALE-HOPPER—A young man or a young woman who makes a business of appearing late at dances after the ticket-takers have gone.
PUNK—Any sort of an undesirable person of either sex.
SCANDAL-WALKER—A college boy or girl, so called from their manner of walking and dancing.
HOLAHOLY—A girl or boy who objects to necking.
DUMBDORA—A stupid girl.
JANE—A girl who meets you on the stoop.
NICE GIRL—One who takes you in and introduces you to her family.
GLIMMERS—The eyes of either sex. “To put the glimmers on”—to take notice.
A BROOKSY BOY—A good dresser, or an overdressed young man.
THE CAT’S PAJAMAS—Anything that is very good.
HOT-DOG—A joyous expression of approval.
GIVEN THE AIR—When a girl or fellow is thrown down on a date.
GERRYFLAPPER—A girl who thinks she looks like Geraldine Farrar ³ and adores her.
FINAGLER—A young man who stalls until some one else pays the check.
LOLLYGAGGER—A young man addicted to attempts at hallway spooning.
SMUDGER—One who does all the close-fitting dancing steps.
DID I WAS—An exclamation of approval.
WIND-SUCKER—Any person given to boasting.
LOW-LID—The opposite of highbrow.
LENS-LOUSE—A person given to monopolizing conversation.
HIS TEMPO’S BAD—A phrase used about any one off color in any way.
GREASE BALL—A foreign cake-eater.
NECKER—A person of either sex given to cheek to cheek dancing.
BUN-DUSTER—Another name for a cake-eater.
RUG-HOPPER—A young man who never takes a girl out. A parlor-hound.
CLUCK—A girl who dances clumsily.
DEW-DROPPER—A young man who does not work, sleeps all day and gets up at 6 p.m.
PRINCESS MARY—Any girl who is soon to be married.
OUT ON PAROLE—A person of either sex who has been divorced.
SUB-CHASER—A man who tries to pick up girls on the street.
TWIST—A girl one takes out to dances, “to twist and twirl.”
STORM AND STRIFE—A married cake-eater’s way of referring to his wife.
BARLOW—A girl, a flapper, a chicken.
BLOUSE—To leave it, to beat it, to meander, to take the air, to blow. “Let’s blouse.”
GOOFY—To be in love with or attracted to. “I’m goofy about Jack.”
APPLE SAUCE—The same as “Blaah”—meaning there’s nothing to it. It’s out.
KIBITZER—A fellow who stands around a dance hall or card game giving unasked for advice, a noisy wall flower.
A WALLIE—A goof with patent-leather hair.
DUMB-BELL—A somewhat archaic expression indicating a person who has more looks than brains. Also, dumb-bunny.
JAMMED—Intoxicated, bolognied, pie-eyed, piffled, shot, shellacked, canned, out like a light, stewed to the hat, potted, jiggered, tanked.
GOOF—A fellow, a sap, a guy, a fish, a bird.
THEY—Used with tones of disgust, the older generation. Objecting parents.
BELL POLISHER—A young man addicted to lingering in thes [sic] vestibule at 1 a.m.
BOFFOS—Dollars, otherwise known at [sic] rocks, chips, seeds, berries and jack.
PIPE DOWN—A synonym of “lay off”—to hush up.
CRASHER—Any one who comes to parties uninvited. Crashing-party, where several young men in a group go to places uninvited.
BLAAH—Anything that is no good, or is “out.”
TOMATO—A pretty girl who can dance like a streak but is otherwise a dumb-dora.
SCOOT—A term used by girls to denote a youth who runs an elevator.
WRINKLE—The mother of a stage girl. For example, “Was the wrinkle with her?”

³ Geraldine Farrar (1882-1967) was a U.S. opera singer and film actress.