Daniel, leading up to a kanji lesson which sprouts a beer review:

無 negates anything it precedes, as do 非 and 不. I’m embarrassed to say I can’t remember the difference between the three; whatever – just keep that in mind whenever you see one of those three, okay?

Not okay! Let's talk about it!

Admitting cheerfully in advance that no rule could possibly apply neatly to the entire multi-millenial history of these characters, I would put it this way:

1. 無 = absence (of a quality or thing)
2. 不 = failure (to achieve an implied goal)
3. 非 = non-identity (often negative)

無 is found in negative words like 無知 ("no-wisdom") and 無礼 ("no-manners"), positive words like 無事 ("no-thing, no-incident" = safe) and 無垢 ("no-dirt" = pure), and neutral words like 無機 ("no-life" = inorganic) and 無糖 ("no-sugar").

不 is found mostly in negative words like 不利 ("no-benefit" = disadvantage), 不治 ("no-cure" = incurable). It's also associated with -zu, the classical auxiliary verb of negation; 不 modifying a verb in kanbun invariably ends up -zu or some conjugation of same.

Which is an important point, actually: generally speaking, 無 modifies nouns and 不 verbs, although owing to the way classical Chinese worked the distinction isn't always as clear-cut as it could be. In fact, some concepts appear both ways: 不二 and 無二 both mean "no-second" (i.e. peerless). Note also that the opposite of 無 is 有, meaning "presence (of a quality or thing)", but 不 doesn't really have an opposite because it is not an absolute... although there are apparent exceptions, of course, like 不利 ⇔ 有利.

非 appears in words like 非常 ("not-normal" = emergency, unusual), 非公開 ("not-publicly-open" = private). That is to say, if Y = 不X means only that Y is or does not X, Y = 非X means that Y is the opposite of X, and usually in a bad way: 常識 means "common sense", but 非常識 means "against common sense" rather than "uncommon sense" or even "lacking common sense". And just as 不 is associated with -zu, 非 is associated with ara.zu, the negative form of aru, "to be". (Which, yes, uses the same -zu.)

One great Sino-Japanese word that uses 非 is "人非人", ninpinin. This kanji palindrome means something like "(a) human inhuman", i.e. a brute in the form of a man, a person who does things no person should countenance. Here's an example from Fukuzawa:

但し天地は広し、真実の父母が銭の為めに娘を売る者さえある世の中ならば、所謂親の威光を以て娘の嫁入を強うる者もあらん。昔の馬鹿侍が酔狂に路傍の小民を手打にすると同様、情け知らずの人非人として世に擯斥せらる可きが故に、斯る極端の場合は之を除き、全体を概して言えば婚姻法の実際に就き女子に大なる不平はなかる可し。

Heaven and earth are wide; in this world in which some parents would sell their own daughter out of avarice, there are no doubt also those who would use their authority (威光) as parents to force their daughters into marriage. Like the idiot samurai of old who cut down commoners on the roadside for fun, such people will be rejected by the world as heartless 人非人; and so excepting these extreme cases, generally speaking the reality of the marriage law imposes no great unfairness on women.

This is the conclusion of a paragraph in which he posits that although parents may propose marriage partners to their children, they are unable to force the issue, and therefore any Westerners who opine that (Meiji) Japanese-style marriage is decided by parents alone are ignorant and ignorable: "父母は唯発案者にして決議者に非ず、之を本人に告げて可否を問い、仮初にも不同心とあらば決して強うるを得ず。直に前議を廃して第二者を探索するの例なれば、外国人などが日本流の婚姻を見て父母の意に成ると言うは、実際を知らざる者の言にして取るに足らず".)