When and how did we start getting "off the dime"?

I believe the meaning has remained the same but reasons changed over the decades. I was told that during the depression if someone saw a dime on the ground they would put their foot on it until they could pick it up. This was frequently in line for the soup kitchen so they would hold up the line and it became a mantra to anyone daydreaming in line. "Hey! Get off the dime!!"

Three early senses of 'get off the dime'

To supplement Josh's answer, I offer these early Google Books matches for "get off the dime," all from the middle 1920s.

From California Engineer (December 1925) [combined snippets]:

GET OFF THE DIME!

Mental indolence is a spectre that haunts every college campus. It is a disease, a disease as insidious as the Black Death of the Middle Ages. It is all the more dangerous because it is extremely difficult to combat, as was the Black Death three hundred years ago. From this disease no one is immune. It is not only the unintelligent and naturally slothful people whom it affects, but the mentally alert and industrious fall prey to its contagion.

From The Stanford Quad (1926) [combined snippets]:

Now comes too soon the final Reckoning-time;

Five Exes in one Quarter is a Crime

I'll have to seminar to beat the Band

Or roll the Stuff; I must get off the Dime.

From The Seventh Regiment Gazette, volume 40 (1926):

In fact, I have a very clear recollection of a baseball game at West Point [New York, home of the U.S. Military Academy], two years ago, when I was, through no fault of my own, an integral part of the battery of "Stricker and Restor for the Seventh Regiment of New York." The clearest recollection of all, however, is of an exciting part of that very same game, with two scurrying Regimental players dashing madly around the bases, with me in hot pursuit, having just made a lucky drive, and with various rooters in the grandstand, of whom even at this very late date, I am undecided as to whether they were with, against, or 'way ahead of me, shouting various verbal forms of either encouragement or discouragement ; I don't know which, such as, "Take that house off your back," "Throw the lead out of your feet," "Come on, get off the dime," and other jocular remarks intended to belittle my ability, ...

From William Morse, "Stanford Expressions," in American Speech, volume 2 (1927):

get off the dime — start (1)

The marker (1) means, according to Morse, "Used at Stanford [University]," as opposed to the marker (2) ("Used on the Pacific Coast") or the marker (3) ("Used on the Pacific Coast and in the Middle West").

Frederick Bolton, Adolescent Education (1931) revisits the expression as follows [combined snippets]:

On every college campus slang is different. The Stanford student says he got a valentine in every course for the simple reason that he didn't "hit a book."

The only solution for the Californian is to "get off the dime" and quit "fussing the co-eds." This slang is so strange that a student traveling from one college to another must almost learn a new language.

These early instances are striking in their distribution: The instance in California Engineer from late 1925 refers to indolence on "every college campus" but obviously is a California publication. This is followed by two seemingly independent reports of the term from sources at Stanford University (in California) from 1926 and 1927, and then by a mention of the expression at Stanford University that very likely relies on the 1927 glossary item.

The geographical oddball is the item from New York in 1926, in which the author recalls the expression being used by baseball fans in New York two years earlier. Arguably, too, although the sense of the expression might be translated as "get off you duff" in both the California college instances and the New York baseball instance, the underlying sense of the phrase is somewhat different: the California instances mean"stop goofing off, and get to work," while the New York baseball instance could be translated as "run as fast as you can, slowpoke."

J.E. Lighter, The Random House Historical Dictionary of American Slang (1993) gives separate entries to the dancehall meaning of "get off the dime" (cited in Josh's answer) and the collegiate meaning:

get off the dime, 1. to move from a stationary position.—used esp. of dancers in a dancehall. [Cited examples, starting with the 1925 Van Vechten quotation that Josh's answer includes, omitted] 2. to take action after a period of indecision or procrastination; to act. [Cited examples, starting with the 1926 American Speech glossary mentioned earlier in this answer, omitted.]

Lighter puts the dancehall meaning first because the Van Vechten novel (set in New York City) came out a year before Morris's "Stanford Expressions" article did. But as noted above, attested use of the procrastination sense of "get off the dime" in California goes back to December 1925.

So we have three distinct senses of the same expression emerging almost simultaneously on opposite coasts of the United States, two of them first attested in 1925 and the other attested in 1926.

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Update November 19, 2020: A 1924 newspaper item on 'get off the dime'

An Elephind search turns up this interesting ad slightly earlier treatment of "get off the dime" from "'Get Off the Dime: New Coinage for Dance Places" in the New York Clipper (April 10, 1924):

"Git off the dime" is the new expression adopted by floor managers in the dance places of the more humble sort when they admonish couples who wiggle or keep too close together while stepping.

It used to be "break it up."

Spielers getting "git off the dime" know what it means and obey the command or get the "rush" and the air.

This instance ads considerable weight to the dance-hall origin theory; more particularly, it emphasizes the expression as bouncer jargon—and it places the term's origin in New York City. It does not, however, speak to the thinking behind the reference to "dime"—although it is certainly compatible with the notion of associating "dime" with someone who pays a dime to dance with a partner-for-hire.

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'Play on a dime'

But there is a further complication. Lighter includes an entry for a different expression involving dimes that was in use by 1929:

play on a dime Baseball. ...[First cited use:] 1929 N.Y. Times (June 2) IX 2: A fielder who fails to cover much ground is said to be "playing on a dime."

A Google Books search turns up an instance of this usage from The Literary Digest, volume 114 (1932):

Another early revolutionist, Charlie Comiskey, was first-baseman with the old St. Louis Browns, and those were the days when an infielder, hugging his bag, "played on a dime." Charlie went to the Browns' manager, and said:

"See here, boss! I wish you'd let me play off that bag a bit. I could cover more ground, and I could hustle back in plenty of time to take the throws after the ball was hit. Want to try it out some day?"

These entries involving "play on a dime" as baseball lingo for guarding a base closely on defense instead of playing off the base mesh nicely with the New York baseball recollection in which fans yell at a runner to "get off the dime."

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'Stand on a dime'

The only remaining point I want to raise here is that the expression "stand on a dime" shows up in a sporting context as early as 1916, although the sport is boxing. From Fred Hunter, "The Hypodermic Needle, in the Omaha [Nebraska] Bee (March 12, 1916):

After losing a ten-round bout to the champ, Ad Wolgast demands the attention of the public by declaring "I can beat Freddie Welsh in a twenty-round fight." Probably, Ad, probably, if Eddie Rickenbacker trains you and you make Freddie stand on a dime.

And from the [Paris, Kentucky] Bourbon News (December 29, 1922) this space-filler joke:

If some fellows would stand on a dime they would remind us of the Woolworth stores—nothing over ten cents.

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A bold but iffy hypothesis

On the available evidence (which is pretty skimpy) I offer this hypothesis: (1) "Stand on a dime" (remain stationary or motionless) comes into use by 1916 as a way to describe standing still. (2) In baseball, by the early 1920s (though I have no corroborating evidence for this date, as the earliest documented use of the expression is from 1929), the tactic of guarding a base by standing on it until the batter hits the ball gives rise to the expression "play on a dime." (3) By 1924, when a baseball player needs to run fast, fans sometimes shout "get off the dime." (4) Through the national mobility of baseball and baseball slang, "get off the dime" by 1925 reaches California, where, on college campuses, it acquires the meaning "do nothing, procrastinate, behave indolently." (5) Either from the baseball sense of "get off the dime" or by an independent route from the old "stand on a dime," dancehall slang by 1925 adopts "get off the dime" as a way of saying "move from a stationary position [on the dancefloor]."

There are several holes in this hypothesis, starting with the big ones at steps 2 and 4; but whether it turns out to be valid or not, I would love to see additional data on early instances of the three expressions discussed above—"stand on a dime," "play on a dime," and "get off the dime"—as well as on the relative age of the sporting, collegiate, and dancehall usages of "get off the dime."

Aside from two possible but unlikely glancing references in ads (05 June 1902 and 27 June 1902) to the set phrase 'stop on a dime', I found no instances of that phrase prior to the first instances of 'get off that dime' in 1907 and 1914 (shown following). I also did not find earlier uses of 'stop on a dime' with 'dime' word variants 'penny', 'pence', and 'sixpence'; neither did I find earlier uses of the potential transitional phrase variants 'stand on a dime' and 'play on a dime'.

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"Get off that dime" and order your job work at the Register Office where you are sure of getting satisfaction and prompt attention.

Cortland Register (Courtland, Kansas), 12 Jul 1907, p. 5 (paywalled).

This first, 1907 use, might be interpreted as something other than a reference to a set phrase, and it might have a dual meaning: the "dime" in question might be one the Register wants to pry from the grip of customers, and it might be part of the set phrase, as suggested by the surrounding quotes, exhorting those same customers to get moving on a printing job they had been putting off. The two meanings are not mutually exclusive, and this first instance might represent earlier uses in advertising of 'get off the dime' in the sense of "let go of the dime".

Of the proposed earlier sense, "let go of the dime", I have no evidence other than frequent earlier uses of 'stop at a dime', first with reference to market gambles on mining stock costing a dime a share, and later more generally. So 'stop at a dime' appears first (before 1900), followed by 'get off that dime' (1907), which is followed by 'stop on a dime' (1919). Of these, 'stop at a dime' and 'stop on a dime' seem to be related; 'get off that dime', however, does not appear to be related to either of the "stop" phrases.

A second early instance of 'get off that dime' appears in the 1914 Northwestern University (Chicago) year book as the entirety of a pharmacy student's personal slogan.

The use of "that dime" in both these early uses conveys a sense of urgency and immediacy that the weaker "the dime" does not.

The 1907 appearance of 'get off that dime' contradicts the common theories ascribing the origin of the phrase to taxi dance hall parlance, and even the 1914 appearance stretches such theories beyond likeliness. Taxi dance halls, according to the histories I examined (Streetswing.com, Atas Obscura, Wikipedia, The Taxi Dance Hall, etc.), were an innovation in San Francisco in 1913. Nonetheless, taxi dance hall exhortations may well have and almost certainly did contribute to the later development and general adoption of 'get off that dime'.

Early appearances of 'turn on a sixpence' in 1891 (left column, ll. 5-6 below figure) and its US fraternal twin 'turn on a dime' in 1893 (center column, second paragraph from bottom), probably have no closer relation to 'get off that dime' than is implied by the typification of the dime and sixpence as very small flat objects. At any rate, I discovered no transitional variations, discounting the possible but unlikely slantwise references to 'stop on a dime' mentioned in my first paragraph.

The principal candidate transitional phrase, 'stop on a dime', did not appear until 1919, after the 1907 and 1914 appearances of 'get off that dime'. In 1919, then, 'stop on a dime' seems to be an adaptation to the automobile age of the earlier equestrian praise, 'turn on a dime', and it doesn't have any discernible connection to 'get off that dime':

A slight accident occurred on Main Street the other night in which the Ford came out second best. Had it been equipped with the outside brakes which are sold at Smith's Garage, the accident would not have happened, because you can stop on a dime with those brakes. — Adv.

The Evening Kansan-Republican (Newton, Kansas), 18 Oct 1919, p. 2 (paywalled).