From The Phrase Finder:
“…Given that playing tricks on the simple-minded must have been happening since Adam was a lad, it seems odd that ‘fool’s errand’ didn’t emerge into the language until the 18th century. The reason for this is that medieval England had a different name for the sport, which was a ‘sleeveless errand’. From the Tudor era to around the 1700s, ‘sleeveless’ was very commonly used to mean ‘futile’ or ‘trifling’. ‘Sleeveless answers’ were those that gave no useful information and a ‘sleeveless errand’ was a fool’s errand, often used to get someone out of the way. The historian Raphael Holinshed used the expression in Chronicles, 1577:
So as all men might thinke that his prince made small account of him, to send him on such a slevelesse errand.
‘Sleeveless’ had also been used for centuries before with the same meaning as now, that is, ‘without sleeves’, so it’s reasonable to assume that’s where the ‘futile’ meaning of sleeveless derived. What’s not clear, and despite my best efforts I’ve not been able to find out, is why ‘sleeveless’ was used with that meaning. Such usage of the word has long since died out and, although it’s not difficult to make guesses at the link between ‘sleeveless’ and ‘futile’, to know the real truth of that derivation we may need to get aboard a time machine.”
From: The Life and Death of Richard the Third (Act 3, Scene 4) 1633, by William Shakespeare:
My lord of Ely!
BISHOP OF ELY
When I was last in Holborn,
I saw good strawberries in your garden there
I do beseech you send for some of them.
BISHOP OF ELY
Marry, and will, my lord, with all my heart.
Cousin of Buckingham, a word with you.
Drawing him aside…”