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Urban Dictionary was the first result, and in typical Urban Dicitonary fashion, there was one useful entry and then some more interesting ones. The useful entry defines the phrase as "A very silly way to say 'Let's get a move on.'" Thanks, I already knew that.
The second entry is a bit more entertaining: "Lame, 50's, white-people way of saying, "Let's get outta here!" Example sentence: "Golly Gee! There's a trouble a comin'! Let's Blow this Popsicle Stand!"
Then there is the great grand third entry. Because I have not been able to find any confident origin story for the phrase, this one has to remain on the table as a possibility:
"Legend has it that this term was first coined in the 1940's by a Jamaican named Antoine Cleo. Antoine Cleo believed that filling the area of a popsicle with deadly radiation, could be used against certain countries as biological warfare. He believed that the radiation had certain brainwashing chemicles inside of it and that placing the popsicles at stands at random areas could allow more people to buy the popsicles, thus was his plan for world domination.
However, his plans were soon foiled when a strange cult called the Kindred Spirits (The Cult leader was Later identified as a man named Dushka Deshvky) blew up all the popsicle stands in America, including killing the perpetrator, Antoine Cleo.
"Lets blow this popsicle stand" was then started as an inside joke between 4 teenagers, then it spread throughout the United States, the phrase from then on meant, "Lets get out of here fast, before something bad happends to us."
Now maybe after hearing this, we can walk away a little wiser about what this phrase means."
Whoever wrote this definition included a source (Evil Minds of the 1900s), which in the world of academia gives it the most credibility (though a search of that works title does not produce any exact results).
Like I said, I was not able to find any confident results for the phrase origin. Yahoo! Answers "Best Answer" brings up the fact that it couldn't have come from earlier than 1924, because that's when popsicles were invented. Another contributor posits that "I think it was the same place that 'pop a squat' originated.'" One person says it comes from SNL's "Ambiguously Gay Duo," which I have never seen and therefore cannot confirm nor deny the truth of that statement. And of course, there is this:
"In the middle of the 20th century, when the back-street abortions were rampant, the FBI in the U.S. mounted an unprecidented crack down on the abortion providers. In an effort to avoid being arrested (or worse), the abortionists took to hiding their programs in ice cream parlours and popsicle stands. Anti-choice terrorists used the phrase to signal their compatriots to leave the store quickly just before the explosion in order to avoid dectection. Oh, and, I have no idea."
Someone brought it up on a Snopes forum, but that degenerated into people just saying they had or had not heard the phrase before and someone saying they thought it was "taco stand." StraightDope has some references to shows on which the phrase was used, but no origins.
Basically, lots of people know of the phrase, but no one has any idea where it comes from. It seems to have been spawned out of thin air, like how people thought maggots were spontaneously generated from rotting meat. So if you know where the phrase came from, please let the internet know, and you could basically become Louis Pasteur.