Green’s Dictionary of Slang

1. a form of sexual ‘game’ (see cit. 1785).

1732		Harlot’s Progress 27: Moll would propose, / Each there should strip off all their Clothes, / [...] / And if the Cole did higher rise, / A Game at Chuck, a burning Shame!
1785, 1788, 1796		Grose Classical Dict. of the Vulgar Tongue n.p.: Burning shame, a lighted candle stuck into the parts of a woman, certainly not intended by nature for a candlestick.
1795		Fortnights Ramble through London 81: The house was infamous, and having been the scene of much wickedness, was deemed disorderly [...] He added, ‘is it not, think you, a burning shame, to suffer such a nuisance?’.
1811		Lex. Balatronicum.
1823		‘Jon Bee’ A Dict. of the Turf, The Ring, The Chase, etc. 19: Burning-shame — practised upon Bodikins, by the Authorities. who station a man with a lighted candle day and night, with the supposed intention of placing it in a socket of a queer kind, as soon as is convenient.
1843		Satirist & Sporting Chron. (Sydney) 8 Apr. 2/4: Lady G.— What has put you out — is it because the cook burnt the candle so long in her bedroom last night [...] Sir G.— Lady Gipps, hear me, it is not for that I am in this mood, though it was a burning shame whoever paid for it.
1862		E. de la Bédollière Londres et les Anglais 313/1: burning sham [sic], chandelle allumée, qui éclaire des orgies.
1869		Sportsman (London) 19 Aug. 2/1: Notes on News [...] Mr Garden.—I put it to the court whether, when people go to bed, they not think merely of going to bed, and let the candle put itself out. (Laughter.) Mr. Brookes.—My lord, it’s a burning shame. (Laughter.) Mr Garden.—After such a remark that you ought to be snuffed out (More laughter.).

2. a nightwatchman placed at the door of a brothel, holding a lantern, even in daylight, to deter people from wandering in and out.

1823		Egan Grose’s Classical Dict. of the Vulgar Tongue n.p.: burning shame Having a watchman placed at the door of a bawdy-house, with a lantern on his staff, in the day-time, to deter persons from going in and out: done by parish-officers to clear the neighbourhood of such sort of houses.
1830		Egan Finish to the Adventures of Tom and Jerry (1889) 194: [note] This mode of expulsion is generally adopted by the Parish Officers to drive the proprietors of Brothels out of the neighbourhood where they exist, when almost every other attempt to abate the nuisance has failed. Numerous rows are the attendant companions upon the Burning Shame.