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The expression ashtray breath designates foul-smelling breath caused by smoking.

This expression occurs, for example, in the following from the Sunday Mirror (London, England) of Sunday 10^th April 1988 [page 23, column 3]:

Sunday Mirror investigation into the perils facing young smokers
When that first kiss is the last
By STEVE BAILEY and STEVE BOGGAN

Girls are puffing their way to the top of Britain’s teenage smoking league. As boys begin to kick the habit under the weight of health warnings, girls are still defiantly lighting up in frightening numbers.
One in three girls is a regular smoker by fifth form and 71 per cent of them fail to give up when they try.
[…]
The risk of death doesn’t deter them. Nor does the cost—or boys turning down their kisses because of “ash-tray breath.”

The expression ashtray breath seems to have been coined in advertisements for Old Golds, a U.S. brand of cigarettes, published in U.S. newspapers in July and August 1931—the first advertisement that I have found is from the Daily News (New York City, New York, USA) of Monday 13^th July 1931 [page 10]:

AVOID “ASH-TRAY” BREATH
SMOKE PURE-TOBACCO
OLD GOLDS

It isn’t good tobacco that spoils the perfect evening . . . by clouding your breath with the odor of stale cigarettes. Lingering and unpleasant cigarette breath comes from the fumes of greasy, ARTIFICIAL flavorings.
OLD GOLD contains no such added substances. It is a PURE-TOBACCO cigarette . . . free of those oily, foreign flavorings that burn into clinging, staining, breath-tainting vapors.
Whether you are a heavy smoker or a light one, you’ll like the Nature-flavored OLD GOLDS. Their clean, sun-ripened tobaccos are like honey to your throat . . . And they leave no objectionable odors either on your breath, your clothing, or in the room.

NOT A COUGH IN A CARLOAD
KEEP KISSABLE . . . NO “ARTIFICIAL FLAVORS” TO TAINT THE BREATH . . . OR STAIN THE TEETH

The next-earliest occurrences of the expression ashtray breath that I have found are as follows, in chronological order—they explicitly or implicitly refer to the advertisements for Old Golds:

1-: From the Daily Kennebec Journal (Augusta, Maine, USA) of Monday 20^th July 1931 [page 6, column 2]:

In boasting a particular brand of cigarettes the super-sales propaganda advises to “keep kissable” and “avoid an ash-tray breath.” Playing up such dangers imminent to the smoker may encourage some impatient victim to “cut it out” altogether. However, sales increase, we understand.

2-: From The Decatur Daily (Decatur, Alabama, USA) of Tuesday 21^st July 1931 [page 4, column 1]:

We have read lots of advertisements, but listen to this headline, “Avoid Ash-Tray Breath.” Now that’s something new in the line of breaths.

3-: From the column The Spur of The Moment, by Edmund J. Kiefer, published in the Buffalo Courier Express (Buffalo, New York, USA) of Thursday 23^rd July 1931 [page 6, column 4]:

AVOID “ASH-TRAY” BREATH.—Adv.
The suggestion is hardly original enough, though, to take anyone’s breath away.

4-: From the column Stray Thoughts, by X. Spence, published in The Bladen Enterprise (Bladen, Nebraska, USA) of Friday 24^th July 1931 [page 4, column 4]:

Now that the gals have all been well informed on how to keep kissable, the men are being lectured on how to avoid ash-tray breath. And when we have all attained perfection along this line, this kissing business should start in earnest.

5-: From Editorial Points, published in The Border Cities Star (Windsor, Ontario, Canada) of Friday 24^th July 1931 [page 4, column 3]:

Day by day cigaret smoking is becoming more and more complicated. In addition to all the other “don’ts,” we now have to avoid “ash-tray” breath!

6-: From Waterbury Day by Day, published in The Waterbury Democrat (Waterbury, Connecticut, USA) of Friday 24^th July 1931 [page 6, column 4]:

Ashtray breath is the latest American menace. Years ago it was tobacco heart.

7-: From The Fence. A Newspaper So Thoroughly Independent It Takes No Stand on Any Issue, published in the El Paso Herald-Post (El Paso, Texas, USA) of Friday 24^th July 1931 [page 6, column 4]:

By DR. B. U. L. CONNER
The Fence editor received a letter from Mary Jane Davis in Las Cruces.
It said: “I live on a farm and am a girl, 17, but the boys won’t notice me. I am a brunette with hazel brown eyes, laughing red lips, and curly locks.
“A girl came from the city recently to spend a week on our farm. She was not so good-looking but she smoked a kind of cigaret that kept her kissable, she said, and made the boys swarm around her.
“Please, doctor, can you tell me where I can get a box of those cigarets.”
Dear Mary Jane: Maybe the fault lies with you. Do you have an ash tray breath?
Remember, even your best friend won’t tell you.