In the phrase "ask away", I've always thought the word "away" does refer to a distance.
Specifically, it refers to the "distance" from the realm of assumable acceptability. For instance, this may refer to questions that seem to be quite "far" from an expected topic (like being "off topic" from a discussion that has already active), or which are quite "far" from the realm of commonly tolerated protocol (such as asking a question about a sensitive subject, or asking a person who isn't the regular recipient of such a question). If a person wants to ask a question that is within the realm of acceptability, then they could presumably ask without needing an invitation. However, in some circumstances, a potential asker might say:
"Can I ask you a question about something else, about another subject?"
(and the response could be...)
"Ask away"
The phrase may also refer to the idea to "proceed" and "go ahead". For instance, with the term "fire away" (referring to shooting a gun, mentioned by some of the other answers), it means that you don't need to limit yourself to some confines/limits about when a person is not supposed to fire. "Go" ahead, and do it. And, don't "come back" whining about things if things don't happen as expected.
Given that my answer has used the words "distance", "far", "off", "within", "proceed", "go", and "come", using the word "away" feels like a quite natural fit to the numerous other spatial terms.
Update edit: added one word to evidence in prior paragraph, and added formatting for easier clarity
8"reminding himself irresistibly" is a very odd phrase. – Tᴚoɯɐuo – 2016-05-26T13:12:52.450
2http://www.thefreedictionary.com/away - Entry #9 : Freely; at will: Fire away! You could omit 'away' and just say ask, but you would lose some meaning and some sense of how receptive the person telling you to ask is to the question. – ColleenV – 2016-05-26T17:36:07.727
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Please see Not so fast! (When should I accept my answer?) By accepting an answer so soon, you make it less likely to receive additional answers, some of which may be better than the one you selected.
– Alan Carmack – 2016-05-27T13:11:04.1271+1 to question, which has generated several different answers that essentially agree with each other. This one simple-seeming word is unexpectedly challenging to analyze. Good job asking a question that turned out to be quite intriguing. – TOOGAM – 2016-05-27T13:35:50.653