Religious rule could anger non-believers over Nature's origins (5)
I believe the answer is:
canon
'religious rule' is the definition.
(I've seen this in another clue)
'believers over nature's origins' is the wordplay.
'believers' becomes 'can' (I am not sure about this - if you are sure you should believe this answer much more).
'origins' indicates taking the first letters.
The initial letters of 'over nature' is 'on'.
'can'+'on'='CANON'
'could anger non' acts as a link.
I am not very happy about this link. Some or all of it may belong to another bit of the clue.
Can you help me to learn more?
(Other definitions for canon that I've seen before include "Such literary works are genuine" , "Ecclesiastical rule" , "General principle; musical round" , "Collection of writing" , "in the ministry" .)