One of the meanings of beg, according to the Oxford English Dictionary, is "to take for granted without warrant." The OED notes that this meaning is most common in the phrase to beg the question, and indeed all of the citations for this meaning are of similar, though not identical, phrases. Ignoring a few duplicates, here are those citations:
1581 W. Charke in A. Nowell et al. True Rep. Disput. E. Campion (1584) iv. sig. F f iij, I say this is still to begge the question.
1680 Bp. G. Burnet Some Passages Life Rochester (1692) 82 This was to assert or beg the thing in Question.
1687 E. Settle Refl. Dryden's Plays 13 Here hee's at his old way of Begging the meaning.
1852 H. Rogers Eclipse of Faith (ed. 2) 251 Many say it is begging the point in dispute.
The etymology of beg is no more helpful—in fact, it's hotly disputed. Some believe that it came from the Old English word bedecian, which in fact means "to beg," but that word has only been found once in all of the surviving Old English literature, and no clear links beyond its meaning and a very slight phonological similarity have been found between it and beg. Others say that beg came, via Old French, from the Latin begardus or beguin, a Christian lay mendicant order known in English as Beghards and Beguines. Either way, there's no clear connection to the meaning of beg used in beg the question.
What we're left with, then, is this: beg in this phrase means something like "to take for granted without warrant," but it only has this meaning in this and very similar phrases. It seems to have acquired this meaning sometime in the late sixteenth century, but how that happened is a mystery.
In common usage, the phrase off you go has a patronising or dismissive connotation.
It is something you say to dismiss a child - Off you go to school now or Off you go and tidy your room.
It is not used in addressing a superior, a customer or in similar situations where you would not wish to be thought of as even slightly rude.
Best Answer
It's a metaphor from the invention of powered aircraft. During the industrial revolution life was changing quickly. Many people didn't understand the new fangled gadgets. Others thought they did and wanted in on making new inventions.
Many crazy ideas got pitched on newsreels but few ever left the ground. One thing everyone picked up on though was that an aircraft that has never flown is unproven. Once it took to the air and landed safely people gained a lot of respect for the design and its inventors.
Eventually "make that fly" or "get it off the ground" became expressions that applied to more than aircraft. So the question is a metaphor. It's asking if the idea will ever be proven to be viable.