The phrase a storm, or a tempest, in a teacup, or in a teapot, and variants, denote a great commotion in a circumscribed circle, or about a matter of small or only local importance.

The following are the earliest occurrences that I have found, in chronological order:

1-: The phrase occurs as part of an extended metaphor in a letter by a person signing themself ‘A Briton’, published in The Public Advertiser (London, England) of Saturday 16^th September 1775 (“the present Embroil in America” designates the War of American Independence (1775-1783), initiated by the Boston Tea Party of 1773, when the tea was thrown overboard from the ships in Boston harbour as a protest against the taxation of the American colonies by the British Government):

Come we now to surely not an uninteresting Question relative to the present Embroil in America. When the Commotion first began in the Western World, to any Thing of a Statesman it would have been nothing more than a Storm in a Tea-cup that could hardly have drowned a Fly. It has unfortunately, by the Bluster of a North-Easter, been swelled into a Hurricane that has threatened the Submersion of a whole Country.

2-: The phrase, therefore, seems to have first been used to minimise the importance of the War of American Independence. This use of the phrase was in turn apparently satirised by The Tea-Tax Tempest, or the Anglo-American Revolution ([Germany]: s.n., 1778), the title of a print attributed to the German engraver Carl Guttenberg (1743-1790); the title also appears in German, as Ungewitter entstanden durch die Auflage auf den Thee in Amerika, and in French, as Orage causé par l’Impôt sur le Thé en Amérique.
This print depicts Father Time using a magic lantern to project the image of a teapot exploding among frightened British troops as American troops advance through the smoke. Figures representing world opinion look on: a Native-American man (America), a black woman (Africa), a woman holding a lantern (Asia), and a woman holding shield and spear (Europe). At the base of the print are two pictures in ovals comparing the War of American Independence to Holland’s auto-da-fé (1560) and Switzerland’s William Tell (1200)—image: Library of Congress:

In fact, the War of American Independence was often symbolised by steaming teapots and smoking bombs. For example, the following woodcut shows a steaming urn about to explode; it is from Tome 1 of Essais historiques et politiques sur les Anglo-Américains (Bruxelles, 1781), by the French lawyer and historian Michel-René Hilliard d’Auberteuil (1751-1789):

3-: From The Morning Chronicle (London, England) of Saturday 16^th January 1808:

The Jamaica Assembly puts on the appearance of great wrath on account of the abolition of the Slave Trade, and seem to threaten us with very serious commotions. We hope they have not forgot that Mr. Burke ¹ once compared a similar rising to a storm in a tea-pot.

¹ This most probably refers to the British man of letters and Whig politician Edmund Burke (1729-1797); however, I have searched in vain for this phrase in Burke’s writings, in particular in the twelve volumes of The Works of the Right Honourable Edmund Burke (London: J. C. Nimmo, 1887).

4-: From Volume II of The Flowers of Wit, or a Choice Collection of Bon Mots, both Antient and Modern; with Biographical and Critical Remarks (London: Printed for Lackington, Allen, and Co., 1814), by the English clergyman, college teacher and writer Henry Kett (1761-1825):

LORD CHANCELLOR THURLOW. ²
[…]
A person came running almost breathless to lord chancellor Thurlow. “My lord,” said he, “I bring you tidings of calamity to the nation, and I do not know how far the direful effects of it may spread to endanger the church and state.”—“What is the matter, man?” said the impatient chancellor. “My lord,” continued this magnifier of political mischief, “a rebellion has broken out.”—“Where, where?” “In the Isle of Man.”—“A rebellion in the Isle of Man,” repeated the vociferous and enraged chancellor; “a tempest in a tea-pot!”
There is a similar idea in Athenæus. ³

² The British lawyer and Tory politician Edward Thurlow (1731-1806) served as Lord High Chancellor of Great Britain from 1778 to 1792.

³ This is an allusion to the following passage from Δειπνοσοφισταί (Deipnosophistaí), an early-3^rd-century-AD Greek work by Athenæus of Naucratis—as translated by Charles Duke Yonge (1812-1891) in The Deipnosophists or Banquet of the Learned of Athenæus (London: Henry G. Bohn, 1854):

Dorion, ridiculing the description of a tempest in the Nautilus of Timotheus, said that he had seen a more formidable storm in a boiling saucepan.

In De Legibus (On the Laws), the Roman statesman, orator and author Marcus Tullius Cicero (106-43 BC) quoted a proverb based on a similar image—the noun simpŭlum denoted a small ladle:

Excitabat enim fluctus in simpulo, ut dicitur, Gratidius, quos post filius eius Marius in Aegaeo excitauit mari.
     translation by Charles Duke Yonge, from The treatises of M. T. Cicero on the nature of the gods; on divination; on fate; on the republic; on the laws; and on standing for the consulship (London: Henry G. Bohn, 1853):
Gratidius was raising a storm in a ladle, as the proverb is, as his son Marius afterwards did in the Ægean sea.

5-: From The Morning Post (London, England) of Saturday 29^th October 1814:

The “Emperor of Elba” has been compared to “a great sea in a calm;” we should have thought him more like what Mr. Burke calls “a tempest in a tea-pot.”

 

The French equivalent phrase is une tempête dans un verre d’eau, literally a tempest in a glass of water. The earliest occurrences that I have found are used to denote the Geneva Revolution of 1782, which was quelled by the professional troops sent by the Kingdom of France, the city-state of Bern and the Kingdom of Sardinia; these occurrences are from:
– Tome 1 of Voyage en Espagne par M^r le Marquis de Langle (1785), by the French author Jean-Marie-Jérôme Fleuriot (1749-1807);
– Tome 1 of Notions claires sur les gouvernemens (Amsterdam, 1787), by the French author Louis-Sébastien Mercier (1740-1814).

Jacques-Pierre Brissot (1754-1793), a leading member of the Girondins during the French Revolution, used the phrase in Précis pour J. P. Brissot ; contre M. Bexon, se disant représentant de la municipalité de Remiremont (Paris: De l’Imprimerie du Patriote François, 20 septembre 1790), about a publication denouncing the secular canonesses of the Abbey of Remiremont, Vosges, France:

Je publiai ; je ne m’attendois pas au fracas que cette publication occasionneroit dans la cité de Remiremont. A la vérité, c’etoit une tempête dans un verre d’eau, comme on l’a dit des troubles de Genève ; mais c’étoit une tempête excitée par des femmes, elle fut donc violente.
     translation:
I published; I did not anticipate the fracas that this publication would cause in the city of Remiremont. To tell the truth, it was a tempest in a glass of water, as has been said of the disturbances in Geneva; but it was a tempest incited by women, it was therefore violent.