In all probability, sleeves were among the first places for magicians, gamblers and thieves to hide their unmentionables. This method of holding out is possibly older than the printing press, making it difficult to trace a precise origin (although this is strictly conjecture). By 1543, sleeving had become advanced enough for Pietro Aretino to describe a metallic holdout to switch cards from within a sleeve. This was published in Dialogo di Pietro Aretino nel qvale si parla del gioco con moralità piacevole, p. 378. Translations of the gambling and card magic material from this book — along with copious notes — were provided by Aurelio Paviato and Lori Pieper for inclusion in Gibecière, Vol. 2 No. 1, Winter 2007, pp. 85-118.
In the eighteenth century, performers were getting bolder with their sleeving techniques; instead of hiding coins in their own sleeves, they'd use the sleeves of spectators. This can be seen in Richard Neve's The Merry Companion, or Delights for the Ingenious, 1716, p. 90. In the same book, Neve teaches readers how to change a playing card into a live bird: the card goes into the sleeve, the bird comes out. This appears on p. 125.
By the nineteenth century, sleeving had become ubiquitous to the point of magicians feeling the need to point out the fact they weren't doing it. For example, in the anonymous Parlour Magic, 1838, p. 134, the author provides the following script as part of the patter, “'Many persons,' you observe, 'perform this feat, by letting the sixpence fall into their sleeve; but to convince you that I shall not have recourse to any such mean deception; I will turn up my cuffs.'”
While early sleeving techniques relied on gravity or manually placing the object inside, a propelled sleeving technique was described in William Cremer's The Sociable, or One Thousand and One Home Amusements, 1858, p. 374. Propelled sleeving caught on and became prevalent in the twentieth century. J.B. Bobo published several versions in Modern Coin Magic, 1952, pp. 101-103, 107, from magicians including Ross Bertram, Stewart Judah and Dr. E.M. Roberts.