I’m learning French with Duolingo and had the sentence:
“We are humans, not gods.”
It was translated as:
“Nous sommes des hommes, non des dieux.”
I would have translated it as:
“Nous sommes des hommes, pas des dieux.”
Just like I’d translate “These are chairs, not tables.” as “Ce sont des chaises, pas des tables.”
Why “non” in this context? Is it a special rule concerning persons?
The Duolingo forums agree with you.
Don’t assume that Duolingo is 100% correct. I was using it from Spanish to French and often the Spanish would be incorrect.
Definitely no special rule concerning persons.
Don’t forget that by itself “pas”, strictly speaking (or maybe even old-fashionably?), does not indicate negation, which explains why it occurs in expressions like “ne… pas”. “Pas des dieux” sounds more like colloquial or informal language (but don’t hold me to that!)
Native French speaker here… and as usual I don’t know the rule, but “non des dieux” is definitely correct.
I think that you can use the “non” in exactly this type of sentences, when you state that something has a certain nature, and not another. Another example would be:
“Je voulais du pain, non de la brioche” (I wanted bread, not cake)
You can actually use “non pas”, i.e.: “Nous sommes des hommes, non pas des dieux.”
So using “non”, “pas” or “non pas” are all valid options?
It’s correct, but it sounds exceedingly weird if you don’t add an “et” in there. “Nous sommes des hommes et non des dieux” is fine, but “, non des dieux” sounds all foofarah poetic and 16th century.
The more common/fluent form of the phrase would indeed be “pas”. Any form with “non” would be awfully stuffy and old-fashioned.
If you want some rules, such as there be, there are tons of examples in “Le Bon Usage”. Here is a sneak peek at §1029 (Rôle de la coordination), (b) (On oppose deux termes, en repoussant l’un):
A marginal paragraph tells us that
but proceeds to give examples using “point”, “et point”, and “pas” instead, e.g., “Elle ressemble fort à son premier portrait, ET POINT du tout à l’autre (Louis XIV, lettre du 4 nov. 1696)”.
So you can see how crystal-clear it all was, even in the 17th century.
Sounds like choreography for minor deities, to me.
So for example: Il vient de Lens, pas de Calais.
Queue the tired old joke, “I was planning on spending my holidays in Calais, but when I got there the signs all said Pas-De-Calais, so I turned back”
I studied French in High School (and lived in France for most of a year) but don’t remember ever learning this usage of ‘non.’
However, the form with ‘pas’ sounds wrong to me! (though I can’t explain why.) I would reword the sentence to avoid the problem. 
“pas des dieux” is correct. That’s the way we say in the spoken langage. I see “non des dieux” as a translation of English.
Both “…pas des dieux” and …“non des dieux” will be understood by anyone, the latter is slightly more literary. The use of “… non pas des dieux” would probably be used to underline the tempo, as by a lawyer in court etc.