I've deployed this joke periodically since I discovered it; a couple weeks ago I reduced a colleague to severe anger, to my extreme delight. Said colleague (I would say friend, but I suspect no more) asked if the joke was born of the financial crisis and Greek economic collapse, and though I was certain it was older than that, I wasn't certain how much older.
So: I investigated this for you. Google Books was of course my first stop. What turned out to be more interesting than I expected is the way the joke slowly mutates over time, its form undergoing alterations.
The joke is first recorded there in 1929, 110 years after Keats wrote his famous poem.
This hit is from Air Travel News, on a page of miscellaneous jokes (other winners include "The gnu is fast disappearing. Shall this little animal be allowed to become extinct? Gno! Gnever!"), but you'll see that it cites it to Life magazine. Google Books includes that it was in Life in the same year, but Google Books won't let me see those back issues.
It seems to get recycled around those kinds of page-filling joke lists a lot, but the answer begins to mutate. In The Literary Digest in 1936, the answer is "Very little" (plus the joke is cited to The Weekly of Auckland, New Zealand). This answer also appears in The Unitarian Register in 1937 and Typo Graphic in 1940. Some attribute names to the speakers, like "Joe" and "Sam," but in a 1941 issue of Foreign Service (the magazine of the VFW), it becomes a smart-arsed pupil, who says, "It depends on what he does," and this version has some staying power, being the almost universal form during the 1940s.
By 1941, it's common enough to yield a metajoke in the Saturday Review:
I wish I could see more of this one to understand the context better, but it appears to be a series of updates on proverbial boys?
In the 1950s, the smart-arsed student version is replaced with a variant featuring a dumb answerer: someone asks, "What's a Grecian urn?" usually a child in the form, "What's a Grecian urn, daddy?" and the other person says, "I dunno. I guess it depends on what he does."
In the 1960s, it actually becomes a joke by union members to make fun of their own, appearing in two union magazines (Carpenter in 1966 and News and Views by the Ohio AFL-CIO in 1967) in the form, "The youngster, doing his homework, asked his business agent father: 'Dad, what's a Grecian urn?' 'I dunno,' replied Dad. 'It'd depend on what his classification was.'"
In the 1970s, it finally mutates into the form I know, where the answer is just a simple amount of money: "Two-fifty an hour" (Westways, 1975) or "about ten drachmas an hour" (Walt Disney's Story a Day, 1978) or "about 20 drachmas a week after taxes" (Rajasthan District Gazetteer, 1978) or "about three drachmas a day" (Musical Heritage Review, 1983) or "a few hundred drachmas at a bare minimum" (How Do You Get a Horse out of the Bathtub?, 1983) or "ten drachmas a week" (Everyman's Word Games, 1986). Apparently the income rate of a Grecian is highly variable.
I should note that in a 1978 New York Classified, the joke warps all the back around into a non-joke, where the punchline is that there is no punchline: "'My wife just bought a Grecian urn.' 'What's a Grecian urn?' 'You know — "Beauty is truth, truth beauty".'" Yeah, I don't really think it's funny either. The 1970s is also when I found regular hits for "What's a Greek urn?" It does appear one or two times prior to that, but the 1970s is when it takes off; it must be when people who don't actually know the poem get hold of it.
I didn't find any Google Books references using euros, which is how I always tell it. There are a lot of opinion columns about the Greek debt crisis that use it as a headline or an opening line, though.
The joke even gets a mention in "Zero to Hero" in the 1998 Disney animated film Hercules: