I was curious to know whether the "ask not" preface that people commonly attach to Donne's original wording was an artifact of the early 1960s, perhaps under the influence of John Kennedy's "Ask not what your country can do for you" rhetoric in his inaugural address of January 20, 1961, or whether the cobbled-together wording preceded Kennedy's speech. To find out, I ran Google Books searches for the phrase (in various forms) from 1600 through 1961.
Clearing up a misdated match
As noted by FumbleFingers in a comment beneath the posted question, a Google Books search for the now-familiar "Ask not..." form of the phrase returns a very early match allegedly from League of Wisconsin Municipalities, The Municipality (1936):
"No man is an island, apart to himself.....so ask not for whom the bell tolls, it tolls for thee." John Dunne wrote these lines two centuries ago. Yet they embrace us today.
If the 1936 date were correct, this match would antedate both Kennedy's speech and Hemingway's novel, For Whom the Bell Tolls (1940)—whose epigraph, by the way, quotes Donne's poem accurately. But if you combine snippets to extend the boundaries of the viewable text from the "1936" source, you'll find that an earlier paragraph begins with a reference to "President-elect Nixon"—proof that the quoted language is actually from a different and much later source—specifically, from Annual Congress of Cities, volume 45 (1969). The Google Books match from 1936 is based on erroneous linkage between the 1969 text and the title page of an unrelated 1936 periodical collection.
Early confirmed matches for 'Ask not for whom the bell tolls'
I mentioned earlier that the epigraph Hemingway uses for his novel quotes Donne accurately. So does the 1943 film version of the novel (starring Gary Cooper and Ingrid Bergman), according to the IMDB listing for the movie:
Opening credits prologue: any mans death diminishes me, because I am involved in Mankinde: and therefore never send to know For Whom The Bell Tolls It tolls for thee.
Nevertheless, legitimate matches for "ask not for whom the bell tolls" begin popping up soon after Hemingway's novel appeared.
From "Pinkett's Prattle," in the [Lincoln University, Pennsylvania] Lincolnian (March 10, 1941):
Glancing over the headlines, I see that: ... Over 500 persons were arrested in Delaware for Sunday blue law violations. Who arrested the cops? ... Connie Mack rates the Yanks the greatest team of all time. I wonder if anyone had the nerve to ask him what he thought of the A's? ... Eighteen members of Bulgaria's parliament voted against Hitler's occupation of that country. "Ask not for whom the bell tolls; it tolls for them" . . .
From "Exchanges: Toll On, Toll On," in the Vassar [College, Poughkeepsie, New York] Miscellany News (June 7, 1944):
News of servicemen on Campus told by the "Minnesota Daily" includes what they term "Vital Statistics: budding leaves, sprouting trees, gentle zephyrs and sunny days are taking their toll. We have it on very reliable authority that Miss Muriel Goldsmith and Pfc. Joe Cantor are on the verge of matrimony."
This use of the word "toll" seems to us unnecessarily lugubrious; its moody connotations are infinite—"Ask not for whom the bell tolls," and "The curfew tolls the knell of parting day." We could continue.
From "Quiz," in the [Perth, Western Australia] Daily News (December 21, 1946):
9—You know the book and film title "For Whom the Bell Tolls." Here is the quotation from which it comes "Ask not for whom the bell tolls. It tolls for thee." Who was the English poet who wrote the poem ? Swinburne; Milton; Shakespeare; Donne; Shelley?
From Richard Boyer, If This Be Treason (1948) [combined snippets]:
...in Spain are in prison. So is the country. Ask not for whom the bell tolls. It tolls for thee.
But it need not. It will not if the people organize for peace and abundance behind the Progressive Party and Henry Wallace. It will not if the people mobilize and force President Truman to live up to his campaign promises; to move for international peace, to liquidate the Taft Hartley Law, to terminate the cold war, to maintain the Bill of Rights for all Americans and not only for these Americans with whom Wall Street agrees.
From Northwestern University on the Air, the Reviewing Stand (1948) [combined snippets]:
MR. HAGSTRUM: I was rather amused to read that after Hemingway's For Whom the Bell Tolls came out with its quotation from John Donne's Devotions people came to the libraries and wanted the complete works of John Donne. Here was one book which influenced another much in the same manner as a movie will influence the sale of the book.
MR. BUCHANAN: That's because of "Ask not for whom the bell tolls. It tolls for thee."
MR. HAGSTRUM: Exactly.
Confirmed matches from the decade 1951–1960
Three of the five matches for "ask not for whom the bell tolls" that I found from the 1940s were from college newspapers, along with one from an Australian newspaper quiz and one from an American political screed. In the 1950s and in 1960, the wording appeared much more frequently (I found sixteen instances of it) and in a wider array of sources.
From Official Proceedings of the New York Railroad Club, volume 62 (1951) [combined snippets]:
You probably remember Ernest Hemingway's book of a few years ago entitled "FOR WHOM THE BELL TOLLS." The theme was that when one man dies, a little bit of all mankind dies. The full quotation on which he based his story was, "Ask not for whom the bell tolls, it tolls for everyone." If anything was ever true as it applies to business that certainly is true today. Years ago the bell tolled for the railroads. No one was very frightened then, but as the bells toll for others we gain understanding and frightened comrades.
From Aircraft, volume 13 (1951) [combined snippets]:
And it is a well known truth that for the first four or five postwar years the little man knew little and cared less about how he could personally be affected by the earlier counterparts of the violation of an obscure place called Korea or the usurping of an oil refinery in Abadan. To him, "ask not for whom the bell tolls, it tolls for thee", was only a gloomy quotation from a poem, not likely to be construed as a warning.
From The Election Issues of 1952: A Series of Addresses and Papers (April 24, 1952) [combined snippets]:
If American liberty is, in fact, being lost, don't blame it on the Constitution, nor on the statute law, nor even on what Whitman calls "the insolence of Elected Persons". The blame will lie, not on the extension of federal power, but on the ordinary citizen who lies supine while he is being stripped of his most valuable possessions. For what they can do to one man, they can do to any other. "Ask not for whom the bells tolls. It tolls for thee."
From New Zealand Engineering, volume 7 (1952):
Hence, to prevent his dying of starvation, or his children of disease and malnutrition, has an appeal to us. The quotation from John Donne, from which Hemingway took the title of his book, "For Whom the Bell Tolls", seems to me to have especial relevance here: ". . . for I am involved in all mankind. Ask not for whom the bell tolls. It tolls for thee.
We are all involved in all mankind.
From "Tower to Town," in the [Boston College, Massachusetts] Heights (November 5, 1954):
There'll be a hot time in the old town tonight (and tomorrow). We mention a few diversions that could succeed in dragging you away from the three r's.
"Ask not for whom the bell tolls; it tolls for thee." A series of PreCana Conferences for engaged couples will begin Monday evening, November Bth, at St. Alphonsus Hall, Smith St., Roxbury. For information on this invaluable aid to a successful marriage, contact Mr. or Mrs. Joseph Hanbury, 88 Calumet St., BE 2-9281. Reservations are necessary.
From Gladys Edwards, Not in the Stars (1945[?]/1956) [combined snippets]:
"Man Is The Brother of Man ..."
The poet spoke a great truth when he said, "a part of all humanity and nothing that is human is alien to me." You have heard it expressed in many ways. An old proverb says, "Man is the brother of man whether he likes it or not." John Donne said, "Ask not for whom the bell tolls, it tolls for thee. " It is said in many ways because, however it is expressed, it is one of the great truths.
From "Addresses by Members of the House of Representatives," in The Lithuanian Situation (1956):
Mr. HOLTZMAN. ... As no man can be an island entire unto himself, so no nation can stand apart and enjoy freedom while others are crushed by tyranny, for every bit of freedom that is lost anywhere in the world makes every man that much less free. In the words of John Donne, ask not "for whom the bell tolls. It tolls for thee," wherever freedom is destroyed.
From Congressional Record: Proceedings and Debates of the ... Congress, volume 102 (1956) [combined snippets]:
Mr. EDMONDSON. Mr. Speaker, it has been a privilege and an honor to serve in the House of Representatives with the gentleman from North Carolina, CHARLES DEANE. There is an old bit of verse which includes the words: Ask not for whom the bell tolls. It tolls for thee.
From Hollington Tong, What is Ahead for China?: Collection of Speeches, June 1956-February 1957 (1957) [snippet view]:
And so, I tell you in the words of the English poet, if that dread combination ever hardens, then ask not for whom the bell in Red China tolls. It will toll for you—for all of us!
From Georgia State AFL-CIO, Proceedings of the Annual Convention (1957):
We are not asking for ourselves. I read it someplace somewhere, and I guess there was a movie made someplace somewhere that they called “A Bell for Adano."
Somebody said in that, I think, "Ask not for whom the bell tolls, for the bell tolls for thee."
And that is why when there is a strike of the Dobb's House workers, when there is a picket for the painters, when there is a picket on for any union, that is your union on fight, that is your union that is trying to live, that is the organization you ought to support.
From Selections from Freedom, volume 6 (1957):
We recognise that as long as we are careless of the well-being of others we provide conditions wherein both they and we suffer. As Eugene Debs said: 'While there is a soul in prison, I am not free' or John Donne: 'Ask not for whom the bell tolls : it tolls for thee.'
From Cooperative Union, Report of the Annual Co-operative Congress (1958):
Ask Not For Whom The Bell Tolls
Mr. A. E. Jupp, C.S.D. (Co-operative Productive Federation): I speak as a delegate from the Co-operative Productive Federation. When I listened to the first few contributions to the discussion I began to wonder whether the Co-operative Movement had given up its celebrated role of self-criticism for one of self-castigation. The punishment meted out for sins of omission in the past made me wonder whether we should be so exhausted by that punishment as to be unable to face the future.
From Punch, volume 236 (1958) [combined snippets]:
Later (by wire):
IF NO REPLY TO MY LAST LETTER WITHIN THREE DAYS SHALL HAVE NO OPTION BUT TO ASSUME IT WAS SENT IN ERROR TO EVENING STANDARD STOP NOW THAT YOU HAVE GOT HAILSHAM DOWN THERE ASK NOT FOR WHOM THE BELL TOLLS STOP WHAT ON EARTH CAN CHELTENHAM AND TUNBRIDGE WELLS BE THINKING OF YOU EXCLAMATION MARK
From Parliamentary Debates (Hansard): House of Representatives, volume 14 (1959) [combined snippets]:
which will not face up to the fact that we have 81,000 registered unemployed and which will not recognize the grave moral effects of unemployment, the cancer in the heart of the community. We are reminded of the old proverb, "Ask not for whom the bell tolls". Unemployment affects not only the worker but every one in the community.
From Frederik Gruber, Aspects of Value (1959) [combined snippets]:
Every individual in the free world wants, for himself at least, freedom, welfare, equality, and peace: freedom both for individual self-realization and from oppressive governmental and organizational control; welfare for the pursuit of happiness and the good life; equality and the recognition of human dignity and worth as a material right from the Creator; and peace within the framework of democratic liberty. Unfortunately, many men are not convinced of the "togetherness" about which Cowan writes, especially in "The Magic Garden," nor, do we realize fully the impact of John Donne's plea for brotherhood in the lines, "Ask not for whom the bell tolls. It tolls for thee," or the neighborliness of the Good Samaritan.
And from a speech of February 1, 1960, in Congressional Record: Proceedings and Debates of the ... Congress, volume 106, part 2 (1960):
NO MAN IS AN ISLAND—ASK NOT FOR WHOM THE BELL TOLLS
No man is an island, entire of itself—
Said John Donne‚ ...
Altogether, I found twenty-one matches for "Ask not for whom the bell tolls" prior to 1961: five from the period 1941–1948 and sixteen from the period 1951–1960. The sources of these matches ranged from college newspaper reporters to speakers at trade associations and union gatherings to political commentators to (of course) elected officials.
Purely on the basis of the time sequence, I suspect that the ultimate cause of the misquotation of Donne's verse was Hemingway's novel—not because it misquoted Donne, but because it popularized a previously rather obscure verse known to relatively few people, and it did so without fully repeating the original wording for context. Presented with the fragment in Hemingway's title, people supplied the "Ask not" prefatory words to round the phrase into a complete sentence in a simple yet solemn (not to say stilted) form.
By 1960, awareness of the expression in its "Ask not" form was widespread. The chronology indicates that it was probably firmly established in popular culture before Kennedy's "Ask not what your country can do for you" inaugural speech of 1961. Indeed, if anything, it may have influenced Kennedy's choice of his syntactically similar phrase.
Ultimately, I couldn't find a single obvious source of the "Ask not for whom the bell tolls" wording that would explain its popular adoption. Rather, it seems to have developed slowly, over a period of years, as people unfamiliar or only slightly familiar with Donne's original wording came to believe that the "Ask not" wording was the original form of the phrase.