The informal phrase to drop somebody or something like a hot potato, and its variants, mean: to abandon a person or undertaking hastily, especially when they become controversial or difficult to handle—as a hot potato is.

This phrase occurs, for example, in Online dating? Think I’ll go to the pub, by Jeremy Clarke, published in The Sunday Telegraph (London, England) of Sunday 12^th February 2012:

Cowgirl and I spent the next three weekends at the spa hotel. Then we agreed we might as well get married and live in a country cottage with roses round the door. I gave notice at the register office. And only then did she realise that although I often wrote articles for the serious papers, I wasn’t a man of substance. She dropped me like a hot potato. It was back to the drawing board.

The earliest occurrences that I have found of the phrase to drop somebody or something like a hot potato and variants are as follows, in chronological order:

1-: From Paddy Hew; A Poem, from the Brain of Timothy Tarpaulin. Whistled by a Sea Lark (London: Printed for Whittingham and Arliss, 1815):

So say no more about the matter,
But drop it—like a hot potatoe.

2-: From a letter to Mr. ******, dated Wednesday 13^th November 1816, published in the Liverpool Mercury. Or Commercial, Literary, and Political Herald (Liverpool, Lancashire, England) of Friday 15^th November 1816:
—two remarks: 1) in the original text, the writer’s signature and several words are illegible (in the transcription, below, a question mark follows each of the words that I have deduced from the context); 2) there is no clue as to the nationality of Mr. ******:

In the Courier of the 16^th ult. in allusion [?] to the sinecure, the mention of which, to use the [?] simile of one of your countrymen, you have “dropped like a hot potato.”

3-: From the proceedings of the Carlisle Convention of Independent Republican Conventionalists, reprinted from the American Volunteer (Carlisle, Pennsylvania, USA), published in The Republican Compiler (Gettysburg, Pennsylvania, USA) of Wednesday 22^nd March 1820—“dropp’d” and “like a hot potato” are in italics in the original text:

The nomination having been gone through, fifteen candidates appeared on the list; two of the members got up and begged leave to withdraw the names of those those [sic] they had nominated! Mr. Fulweiler was one of them; but Mr. Mayer, who acted in a double capacity, as both chairman and secretary, objected to this mode of proceeding. It was then agreed to take the sense of the meeting, by calling over the names of the members, and asking each who he voted for? John M. Taylor, of the city of Philadelphia, being called on; to the astonishment of all present, dropp’d his man, John Steele, like a hot potato, and voted for Gen. Joseph Hiester!

4-: From a letter to the Editors of the Penny Cat-call, by one Egerton Smith, published in The Kaleidoscope; Or, Literary and Scientific Mirror (Liverpool, Lancashire, England) of Tuesday 7^th August 1821—here, Pat, a shortened form of the male forename Patrick, denotes an Irishman:

You have a very convenient knack of overlooking those facts which it does not suit your purpose to examine. Thus you altogether pass over the inconsistency with which you stand taxed of blowing hot and cold, especially with regard to Mr. Bass: you prudently drop that subject, as Pat says, “like a hot potato.”

5-: From the chapter on Belfast, in Ireland exhibited to England, in a political and moral survey of her population, and in a statistical and scenographic tour of certain districts (London: Baldwin, Cradock, and Joy, 1823), by ‘A. Atkinson, Esq., late of Dublin’:

This town, within the last ten years, has been reputed to contain nearly 4,000 houses (inclusive of the suburbs), and a population of 30,000 souls, which, with the exception of a few private families and professional individuals, is composed of persons in the numerous walks of trade; all of whom, from the brogue maker to the banker, stand, like the men of England, on a level with their business, attend assiduously to it, and seem to think of nothing else, until the sabbath-day (which they hold in great veneration) arrives, when they drop, like a hot potatoe, all the small concerns of this lower world, and assemble, shaved and white-washed, some to chaunt a Latin hymn to the music of the holy Pope; others, a psalm in English, to the more modern notes of the English bishops; others to the most barbarous broad Scotch slang that ever disgraced a conventicle in the mountains.

6-: From Cobbett, the Champion of Slavery, published in The Liverpool Mercury. Or Commercial, Literary, and Political Herald, and Lancashire General Advertiser (Liverpool, Lancashire, England) of Friday 26^th December 1823—William Cobbett (1763-1835) was an English author and political reformer; he founded the periodical Cobbett’s Political Register in 1802:

Corporal Cobbett, for the last week, has been “marking time,” but whether he means to come to a dead halt, or to make another charge upon the Abolitionists of Slavery, is a matter of equal uncertainty and indifference; at least to us. If he should drop the subject, as Pat says, “like a hot potatoe,” several probable causes may be surmised, to explain the phenomenon.

7-: From Recollections of an eventful life, chiefly passed in the Army (Glasgow: Published by W. R. McPhun, 1824), by the Scottish soldier and author Joseph Donaldson (1794-1830):

I procured leave from the officer to go to the river for water; intending to proceed a little farther down, to see if I could find anything that I could eat. Turning round the hill, I came to a mill: and, entering it, found a number of soldiers belonging to different regiments of the division busy grinding Indian corn; others were employed drawing a baking of bread, which the French had left in their hurry, when we took up our position. I attempted to help myself to some corn, which was lying in a basket.—“Drop that like a hot potato,” said one of the Connaught Rangers. I tried another basket, but it also was appropriated; and, as there were none of my regiment there, I could not expect to succeed by force; so I left the place, sorrowful enough, on my way back to the picquet, with a cargo of cold water—poor cheer, certainly.