Is there any rule which dictates the ordering of non-proper, non-pronoun nouns in a list?

While there is no rule for ordering general lists, we can probably consider specific cases:
(A) cases where an ordering may be mandatory (eg listing Royalty or Government Officials or by some sort of seniority)
(B) cases where an ordering may be natural (eg ordering temporally or by some measurement like size or age)
(C) cases where phrases may have a standard ordering (eg "Red, White, and Blue" or Binomials or examples here, thanks to @Christian Bouwense & @sumelic)
(D) cases where the writer wants to highlight the most important elements before the least important elements (this is subjective and no rule exists)

Your example of "Design, Operation, and Management" is "temporal order", because:
"Design" of some equipment comes first ;
"Operation" of that equipment comes next ;
"Management" of the systems using that equipment comes last, because we can talk about that only when the first two are ready.
It will probably be a title of a text book.

I have never heard of an official rule, but people usually go by what sounds aesthetically pleasing. There are some combinations that are generally laid out in a set sequence, such as

Red, White, and Blue

if you are describing the American flag, for instance. But in the general case, it is up to you.