The phrase tall poppy designates a person who is conspicuously successful and whose success frequently attracts envious hostility.

This phrase probably alludes to the following story told by the Roman historian Livy (Titus Livius – 59 BC-AD 17) in The History of Rome:
Unable to capture the city of Gabii by assault, Tarquin the Proud (Lucius Tarquinius Superbus – reigned circa 534-circa 510 BC), the semi-legendary last king of Rome, arranged that Sextus, one of his sons, pretending to have escaped from the murderous violence of his father, flee as a refugee to Gabii. When at last Sextus was chosen by Gabii as commander in the war against Rome, he sent a messenger to his father to inquire how he should deliver the city into his hands; Tarquin the Proud silently demonstrated how to deal with the enemies by cutting off the heads of the tallest poppies in his garden:

     from The History of Rome, translation by Rev. Canon Roberts (London – 1905):
When he [= Sextus] thought himself strong enough to succeed in anything that he might attempt, he sent one of his friends to his father at Rome to ask what he wished him to do now that the gods had given him sole and absolute power in Gabii. To this messenger no verbal reply was given, because, I believe, he mistrusted him. The king went into the palace-garden, deep in thought, his son’s messenger following him. As he walked along in silence it is said that he struck off the tallest poppy-heads with his stick. Tired of asking and waiting for an answer, and feeling his mission to be a failure, the messenger returned to Gabii, and reported what he had said and seen, adding that the king, whether through temper or personal aversion or the arrogance which was natural to him, had not uttered a single word. When it had become clear to Sextus what his father meant him to understand by his mysterious silent action, he proceeded to get rid of the foremost men of the State by traducing some of them to the people, whilst others fell victims to their own unpopularity. Many were publicly executed, some against whom no plausible charges could be brought were secretly assassinated. Some were allowed to seek safety in flight, or were driven into exile; the property of these as well as of those who had been put to death was distributed in grants and bribes. The gratification felt by each who received a share blunted the sense of the public mischief that was being wrought, until, deprived of all counsel and help, the State of Gabii was surrendered to the Roman king without a single battle.
     Latin text from Ab Urbe Condita (From the Founding of the City):
Itaque postquam satis virium conlectum ad omnes conatus videbat, tum ex suis unum sciscitatum Romam ad patrem mittit quidnam se facere vellet, quando quidem ut omnia unus publice Gabiis posset ei di dedissent. Huic nuntio, quia, credo, dubiae fidei videbatur, nihil voce responsum est; rex velut deliberabundus in hortum aedium transit sequente nuntio filii; ibi inambulans tacitus summa papaverum capita dicitur baculo decussisse. Interrogando exspectandoque responsum nuntius fessus, ut re imperfecta, redit Gabios; quae dixerit ipse quaeque viderit refert; seu ira seu odio seu superbia insita ingenio nullam eum vocem emisisse. Sexto ubi quid vellet parens quidve praeciperet tacitis ambagibus patuit, primores civitatis criminando alios apud populum, alios sua ipsos invidia opportunos interemit. Multi palam, quidam in quibus minus speciosa criminatio erat futura clam interfecti. Patuit quibusdam volentibus fuga, aut in exsilium acti sunt, absentiumque bona iuxta atque interemptorum divisui fuere. Largitiones inde praedaeque; et dulcedine privati commodi sensus malorum publicorum adimi, donec orba consilio auxilioque Gabina res regi Romano sine ulla dimicatione in manum traditur.

Allusions to the story told by Livy predate the appearance of the phrase itself; for example, the following from The Dublin Evening Post (Dublin, Ireland) of Saturday 29^th May 1813 shows that the Irish Catholics held Lord Castlereagh¹ personally responsible for the rejection by a parliamentary committee of the House of Commons of the clause of the ‘Catholic Bill’ giving Catholics the power to sit in Parliament:

No man is better than his Lordship in the skill to discipline a majority—and, surely, if he were anxious that the Bill should pass, he would have husbanded his resources—he would have left something to concede to his Right Hon. Friends, the Props of High Church—he would not have dissipated all his strength in the first instance—he would have retained something to resign; but, on the contrary, the very first amendment proposed by the opponents of the Catholics gutted the Bill of any thing that was good in it. […]
[…] Irishmen, eminent Irishmen—why did you admit the enemy into the Camp? Bravo Castlereagh. Powerful and successful Ulysses of High Church—bear off Minerva, for Hector slumbers. Bravo Zophyrus², Babylon totters—Darius thanks thee. Bravo Young Tarquin, the Town’s your own. Now, cut off the tallest Poppies.

¹ Robert Stewart (1769-1822), Viscount Castlereagh, Irish-British statesman
² The story of Zopyrus, a Persian nobleman, follows the same pattern as the above-mentioned story of Sextus. According to the Greek historian Herodotus (circa 484-circa 425 BC), when Babylon revolted against the rule of Darius I (circa 550-486 BC), king of Persia (521-486 BC), Zopyrus cut off his own ears and nose, and defected to the besieged city, claiming that he had been punished by Darius himself. Once he had gained the Babylonians’ trust and become commanderof their army, he opened the gates of Babylon and let the Persians in.

Although the figurative use of tall poppy is now considered chiefly Australian, the earliest instances that I have found are from England, Ireland and the USA.

The earliest is from The Public Ledger, and Daily Advertiser (London) of Tuesday 30^th January 1816:

It appears, according to letters received yesterday from Paris, that alarming disturbances broke out recently in the populous city of Lyons, which were repressed with much difficulty, but not without bloodshed. An Evening Paper, which advocates the cause of Ultra-Royalism, attaches the epithets of “Bonapartists and Federists,” to the persons who excited the riots […]. What is the remedy which the French Government is applying? It sends into banishment the old, mostly worn-out, and rich Leaders of the Revolutionary party; it is cutting down, it is true, the tall poppies, which would soon fall from decay; but the little poppies will escape, which are innumerable, vigorous and sound. It is not Cambaceres³ and persons of the same description, who ought to alarm the Government; their great property, their palaces and their age, would offer a guarantee of their tranquillity. It is the young, dashing and aspiring Officer, who is to be met in every village, and who is communicating his angry and impatient feelings to his friends and associates, that is, or ought to be the real object of alarm.

³ Jean-Jacques Régis de Cambacérès (1753-1824): French statesman during the French Revolution and the First Empire

The second-earliest figurative use of tall poppy that I have found is from The Evening Packet, and Correspondent (Dublin, Ireland) of Saturday 2^nd May 1835:

Fanton Fagan’s Journal.

I think I may spake out now. l am a “Ministerial man.” Why would I be ashamed to own it, since my letters have led the way. Some say the Whigs have turned Radicals, others that the Radicals have turned Whigs; but be that as it is, my opinion is that they have met at the half-way house, and the sartinty [= certainty] of the whole matter is that we are all one. More o’ that to us, agrah⁴.
The Scotch Tail didn’t come off second best in this scramble; there are parts of it every where. All the offices, big and little, smell of cane-brimstone⁵; and you can’t lay your hand upon a flower in the Minister’s garden without being stung with a thistle. I hear we are only waiting for our turn, till it gets settled. When Poor Man’s harvest comes, we’ll be saying, as the little potatoes do to the big ones, “Lye over.” And more o’ that to us, agrah.
The Castle of Dublin is to be our first gleaning-field. All the tall poppies that grow in the half-acre must be headed down. Gosset goes it, and dozens of them besides. Lord Fife is set up in Windsor, Mr. Drummond in Dublin, and with such music we’ll drum the rogues out. And more o’ that to us, agrah.

⁴ agrah, also agra and other variants: an Irish-English term of endearment meaning my love, dear
⁵ cane-brimstone: sulphur in rolls or sticks

The following paragraph explicitly associates the phrase to the story told by Livy; it is from The Torch Light (Hagers-Town, Maryland) of Thursday 1^st October 1835:

History tells us that Tarquin the proud, on a certain occasion, cut off, with a stick, the tallest poppies in his garden, and that his son Sextius [sic] followed the example by putting to death the most noble and powerful citizens of Gabii. It will be seen by the proceedings in Frederick Co. that, politically, Mr. Thomas and his friends are imitating the example of Tarquin and Sextius—Indeed it is said some of the tall poppies of our county are in danger of decapitation.

The Australian-English phrase tall-poppy syndrome denotes a tendency to disparage any person who has achieved great prominence or wealth.

The earliest instance that I have found is from Master of his art, a portrait by Jennifer Byrne of James Mollison (born 1931), director of the National Gallery of Australia, published in The Age (Melbourne, Victoria) of Saturday 19^th July 1980:

Explanations are not things James Mollison goes in for, neither in his art nor in his own life. He protects his person ferociously and about the wickedest question one could ask him, worse even than his opinion on ‘Blue Poles’, is where he went to school.
Why? He won’t explain that either, offering only a general observation about the length of Memory Lane (“if I went down I’d never emerge”) and an oblique reference to the tall-poppy syndrome afflicting the Australian media.

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Edit – 21^st April 2020:
Jillian Albrecht, a volunteer transcriber of Trove, the Australian National Library repository of digitised newspapers from Australia, has kindly provided the following extract from the Newcastle Morning Herald and Miners’ Advocate (Newcastle, New South Wales) of Wednesday 31^st May 1944—which contains an early reference to tall poppies:

A Country Safe for Mediocrity

A warning against turning Australia into a country made safe for mediocrity was given by the Bishop of Newcastle (Rt. Rev. F. de Witt Batty) yesterday.
Bishop Batty mentioned that Mr. Philip Arthur Micklem who, in nearly 30 years of ministerial life in Australia had never been called to any of the higher administrative posts, had received high intellectual honours in England.
The Church was not the only department of common life in which the same unhappy tendency was to be observed, the Bishop said.
“It has often been alleged that we, in Australia are apt to be intolerant of exceptional ability that so soon as a man stands out conspicuous in quality above his fellows, there is a tendency amongst the mediocre majority to want to pull him down.
“If this tendency really exists—and I am afraid it does—we ought to set ourselves to correct it without delay. Otherwise we shall run the serious risk of turning Australia into a land of little men, a field in which no poppies rise very much above the average height, a country made safe for mediocrity.
“We are often and rightly reminded of Australia’s crying need for a greater population. We must not forget that of even greater importance than the quantity of a nation’s population is the quality of its leadership.”