A sweating tub used for patients with venereal diseases.
From Green's Dictionary of Slang:
Mother Cornelius’ tub n.
also Cornelian tub, Cornelius’ tub
[a presumed actual Mother Cornelius, whether a nurse or a procuress; but note the masc. ‘Cornelius’ in Taylor, ‘Travels to Bohemia’ (1620): ‘Or had Cornelius but this tub, to drench / His clients that had practis’d too much French’ (i.e. venereal disease); poss. ref. to physician Henry Cornelius Agrippa (1496–1535) a leading advocate of hot baths for medicinal purposes; Henke, in Gutter Life and Language (1988), also notes the possible use of a hard dense wood, necessary to withstand the heavy salt brine used in ‘pickling’ patients, known as cornel-wood, ‘the wood of Cornus mascula, celebrated for its hardness and toughness, whence it was anciently in request for javelins, arrows, etc.’ (OED); also poss. puns on cornel and the ‘cornuted’ cuckold or Lat. cornu, a horn; one’s current incapacity is the result of one’s horn n.2 (1b)]
the sweating tub used in the cure of venereal disease; many ad hoc vars. and extrapolations exist in cits.
I found this via some sources which mentioned "Cornelius tub" in the context of that specific Nashe passage. For example, from this historical paper entitled "Venereal Diseases in Sixteenth-Century England":
In 1561 venereal diseases were treated at St. Thomas's Hospital in the four sweat wards: Job, Lazarus, Judith and Susanna, at the back of the hospital. Berengario Da Carpi (d. 1530), better known as the leading anatomist before Vesalius, was treating the disease by mercurial inunction in 1500 and by fumigation in 1506. That this remained the standard treatment in Britain is proved by the very large number of allusions to it in the literature of the sixteenth and seventeenth centuries. One example is Shakespeare's:
Be a whore still; they love thee not that use thee. Give them diseases, leaving them with their lust. Make use of thy salt hours. Season the slaves For tubs and baths; bring down rose-cheek'd youth To the tub-fast and the diet (Timon of Athens, IV, iii, 83-87).
Another allusion is to 'Cornelius's tub'. How the name came about is unknown. One of the earliest references to it is in Nashe's The Unfortunate Traveller, 1594. 'Mother Cornelius's tub why it was like hell, he that came in to it never came out of it.'
Introductions, Notes and Commentaries to Texts in 'The Dramatic Works of Thomas Dekker' by Cyrus Hoy and Fredson Bowers also contains some other literary references to "Cornelius tubs" and variants thereupon.
'Le Guide de Londres' by F. Colsoni, describing Vauxhall Gardens in 1693, warns its readers against frequenting the prostitutes there and mentions the risk of having to "go to Sweden" or "chez Mr. Cornelius's Tub" if one does so.