When should I include page numbers when citing a specific theorem in a reference?
Here's my answer: page numbers, like much of mathematical writing, follow the "be nice to the reader" rule. This rule says, in effect, "Imagine you are a relatively naive reader of this paper. What information would you like the author to include to help you understand the paper and locate the references?"
Following this rule, if you're citing a result in a long book, you should certainly include the theorem or equation number, and probably also a page number. On the other hand, if you're citing a very short paper with one result, then it's probably not necessary.
As an example of what not to do, take a look at this paper, where the authors write (on page 17, just below equation (30)),
"by a result of Bourbaki , σ-parabolic subsets (respectively, σ-positive systems) of R are just parabolic subsets...
The reference  is to a 300-page book! (To be fair, this was an early version of the paper; in a later version they fixed this.)