There are a couple from the Greek classical period I know about, but I doubt they are the absolute earliest.
The ones I'm thinking of are (1) the story of the lawsuit between Corax and Tisias, with the judge's judgment being "Bad egg from a bad crow," which was a pun on both of their names (Corax = crow, and if I remember it right, Tisias means eggs) and (2) the "is that a scroll in your pocket or are you happy to see me" moment in Plato's Phaedrus, in a dialogue between Socrates and Phaedrus.
Source for the first one is George Kennedy's A New History of Classical Rhetoric, source for the second one is this passage in Plato's Phaedrus:
PHAEDRUS: I will do as I say; but believe me, Socrates, I did not learn the very words—O no; nevertheless I have a general notion of what he said, and will give you a summary of the points in which the lover differed from the non-lover. Let me begin at the beginning.
SOCRATES: Yes, my sweet one; but you must first of all show what you have in your left hand under your cloak, for that roll, as I suspect, is the actual discourse. Now, much as I love you, I would not have you suppose that I am going to have your memory exercised at my expense, if you have Lysias himself here."
hi! you'll find lots on the oldest, and many other jokes in the FAQ
As far as I am aware, the oldest joke is from 1900 BC Sumer and goes like this:
"Something which has never occurred since time immemorial; a young woman did not fart in her husband's lap."
Another very old joke comes from 1600 BC Egypt, attributed to Pharaoh Snofru:
"How do you entertain a bored pharaoh? You sail a boatload of young women dressed only in fishing nets down the Nile and urge the pharaoh to go catch a fish."
The study was conducted by students and faculty at the University of Wolverhampton under the auspices of Dr. Paul McDonald, and was commissioned by the TV channel Dave in an attempt to understand the ancient origins of humor.
http://uk.reuters.com/article/2008/07/31/uk-britain-joke-life-idUKL129052420080731