The phrase paper tiger denotes a person, country, etc., that appears powerful or threatening but is actually weak or ineffective.

This phrase is modelled on Chinese zhǐlǎohǔ, from zhǐ, paper, and lǎohǔ, tiger.

These are the earliest occurrences of the phrase paper tiger that I have found, in chronological order:

1-: From The Chinese: A General Description of the Empire of China and its Inhabitants, published in The Library of Entertaining Knowledge (London: Charles Knight & Co., 1836), by the British colonial governor and sinologist John Francis Davis (1795-1890)—many British and U.S. newspapers reprinted this paragraph, thus popularising the phrase paper tiger:

Some of the ordinary expressions of the Chinese are pointed and sarcastic enough. A blustering, harmless fellow they call “a paper tiger.” When a man values himself overmuch, they compare him to “a rat falling into a scale, and weighing itself.”—Overdoing a thing, they call “a hunchback making a bow.”—A spendthrift they compare to “a rocket,” which goes off at once.—Those who expend their charity on remote objects, but neglect their family, are said to “hang a lantern on a pole, which is seen afar, but gives no light below.”

2-: From the address of the Birmingham Church of England Lay Association, published in The Manchester Courier, and Lancashire General Advertiser (Manchester, Lancashire, England) of Saturday 5^th September 1840:

The weakness of our adversaries has been more and more manifest, and churchmen begin to find, to use a Chinese proverb, that they “have been frightened at a paper tiger.”

3-: From the account of an agrarian insurrection in the state of New York, published in The Morning Chronicle (London, England) of Wednesday 29^th January 1845:

Several men, who were disguised as Indians, are in custody for murder; arms have been seized, and the braggart chief, who adopted the soubriquet of “Big Thunder,” now that he is in prison, shows a pusillanimity and a craven spirit that ill accords with his tremendous threats and name. He has confessed the nature of the conspiracy, impeached accomplices, begs for mercy, and is a complete impersonation of what the Chinese call a “paper-tiger.”

4-: From the Saturday Courier (Philadelphia, Pennsylvania, USA) of Saturday 30^th August 1845—several U.S. newspapers reprinted this essay:

PAPER TIGERS.
“Guns, trumpets, blunderbusses, drums and thunder!”

The Chinese have a very expressive way of describing a mighty blusterer. They call him a “paper tiger.” Nothing could be more significant.—Every neighborhood has its paper tiger; its hero of the hour, an individual who is constantly describing the feats he has performed, or that he would have performed, had he been in such and such a situation. One of the most celebrated poets of our language, has said:
“—the loud laugh betrays the vacant mind.”
[…]
[…] There are “paper tigers” in almost every department of life. “Big Thunder,” the counterfeit Indian Chief, who was recently arrested in New York, charged with murder, may be adverted to as a case in point. While flourishing at the head of his colleagues, and the hundreds of misguided tenantry, he made a most formidable display. He was armed to the teeth, delivered pompous harangues, and threatened terrible things in the way of devastation and death, should any attempts be made to resist him. The Sheriff and one or two men of nerve undertook the awful responsibility, nevertheless,—“Big Thunder” was caught, caged, and in the course of a few days, he was as mild as a sucking dove.
We have some “paper tigers” among the members of Congress. It will not be difficult for our readers to single out a member who has made himself notorious for sound and fury, signifying—nothing! who occasionally makes most terrible predictions, threatens to sever the Union, to declare war, and many other wonderful feats; but whose soaring flights excite no attention whatever, comparatively speaking. The truth is, the member spouts for Bunkum: for his constituents at home, and with little hope of producing effect upon the few Congressional hearers who may at the time be around him.
In general terms, “paper tigers” may be described as that class of persons who are characterised by more sound than substance, who boast of many virtues and attainments, and possess but few—[…] Far better, far nobler, far wiser is the policy—to say less and do more. Deeds, not words—should be the great characteristics of men!

5-: From The Morning Chronicle (London, England) of Tuesday 30^th September 1845:

Philadelphia, Sept. 14.
[From our Correspondent.]

The question of peace or war still hangs in the balance. We have had several arrivals from Mexico, but nothing of a decided character has transpired. I hardly think, however, that the Mexicans can seriously intend war, as time enough has now elapsed to enable the United States government to concentrate a sufficient force by sea and land, on the Mexican coast and frontier, to set all opposition on the part of Mexico at defiance. And perhaps, under the circumstances, this policy will prove to be the very best mode of keeping the peace. True, we have a proclamation by General Arista, in which immediate war is threatened and spoken of as about to be commenced, that the defeat of the Americans is certain, &c. But looking coolly at the matter, this is clearly a brutum fulmen ¹, and Arista appears, in this particular, very much like what the Chinese call a “paper tiger.”

¹ The Latin phrase brutum fulmen (i.e., senseless thunderbolt) designates an ineffective act or empty threat.

The phrase paper tiger gained new currency in the post-war years from its use by the Chinese Communist Party of the USA and other ‘reactionaries’. The following is from an August 1946 interview of Mao Zedong ² with the U.S. journalist Anna Louise Strong (1885-1970), published in Selected Works of Mao Tse-tung—source Marxists Internet Archive:

Strong: Suppose the United States uses the atom bomb? Suppose the United States bombs the Soviet Union from its bases in Iceland, Okinawa and China?
Mao: The atom bomb is a paper tiger which the U.S. reactionaries use to scare people. It looks terrible, but in fact it isn’t. Of course, the atom bomb is a weapon of mass slaughter, but the outcome of a war is decided by the people, not by one or two new types of weapon.
All reactionaries are paper tigers. In appearance, the reactionaries are terrifying, but in reality they are not so powerful. From a long-term point of view, it is not the reactionaries but the people who are really powerful. In Russia, before the February Revolution in 1917, which side was really strong? On the surface the tsar was strong but he was swept away by a single gust of wind in the February Revolution. In the final analysis, the strength in Russia was on the side of the Soviets of Workers, Peasants and Soldiers. The tsar was just a paper tiger. Wasn’t Hitler once considered very strong? But history proved that he was a paper tiger. So was Mussolini, so was Japanese imperialism. On the contrary, the strength of the Soviet Union and of the people in all countries who loved democracy and freedom proved much greater than had been foreseen.
Chiang Kai-shek ³ and his supporters, the U.S. reactionaries, are all paper tigers too. Speaking of U.S. imperialism, people seem to feel that it is terrifically strong. Chinese reactionaries are using the “strength” of the United States to frighten the Chinese people. But it will be proved that the U.S. reactionaries, like all the reactionaries in history, do not have much strength. In the United States there are others who are really strong—the American people.

² The Chinese statesman Mao Zedong (1893-1976) was the Chairman of the Communist Party of the Chinese People’s Republic from 1949 to 1976, and head of state from 1949 to 1959.
³ The Chinese statesman and general Chiang Kai-shek (1887-1975) was the President of China from 1928 to 1931 and from 1943 to 1949, and of Taiwan from 1950 to 1975. He tried to unite China by military means in the 1930s, but was defeated by the Communists after the end of the Second World War. Forced to abandon mainland China in 1949, he set up a separate Nationalist Chinese State in Taiwan.

Likewise, the following is from What the World Is Saying About Us, published in The Philadelphia Inquirer (Philadelphia, Pennsylvania, USA) of Sunday 16^th November 1947:

CHINA
Communist-controlled North Shensi radio, in English Morse to North America, Nov. 7:

“The New China News Agency editorial, entitled ‘A Spark Can Kindle a Wild Fire,’ issued today, commemorates the 30^th anniversary of the Soviet October revolution. The editorial emphasizes that all imperialism and reactionary cliques are nothing but ‘paper tigers,’ only fearful in aspect but actually not to be feared at all. The October revolution proved this point, the editorial states, and the 30 years history following it proves it even more clearly.
“The Chinese people have complete certainty of striking down Chiang Kai-shek’s reactionary clique, despite the aid of American imperialism. The next numbers of years will prove, the editorial concludes, that the Chinese people can create the same mighty achievements as those of the Soviet people in conquering imperialism and domestic reactionaries.”