Do any legitimate conferences charge per publication?
In computer science some legit conferences charge per accepted publication. In my experience, legit but small conferences perform this practice, perhaps to better cover the organization costs. Bigger conferences are often sponsored by big partners, e.g., Google, Microsoft, Facebook, Cisco, .., so they do not need to look for money elsewhere.

Generally speaking, conference registration pages are not very clear on this issue. Let's observe the registration page of the three biggest and most respected software engineering conferences, that are ICSE, FSE, and ESEM (there might be another better example for the third place here).

ICSE 2015 registration page:


  As specified in the acceptance letter, each accepted paper must be “covered” by one registration in the name of one of the authors.


FSE 2014 registration page:


  our paper submission and acceptance implies that one of the paper's authors should register the paper using non-student registration, i.e., either as non-member or ACM/SIGSOFT (Student) member, and present the paper at FSE 2014.


ESEM 2014 registration page:


  Conference registration is compulsory for at least one author of accepted papers and posters.


In my opinion, all three pages might imply that the conferences require a per publication fee. But, 


I have just asked an author who was accepted twice for ICSE, and he was not required to pay twice the registration costs. (confirmed by @Psychonaut)
@Psychonaut can confirm that FSE does also not require multiple registrations.
I have asked some ESEM authors and organizers, and ESEM does not charge twice if a single author presents two papers.


Smaller conferences tend instead to ask for a per publication fee in case of multiple accepted papers and one author is presenting them.

An example is PROFES, which is a small yet respected and established venue.


  Each paper published in the conference proceedings (full/short/poster paper) requires at least one early registration to the main conference as regular attendee.


I happened to be there with two papers (one full, one short), and I had to pay twice.

I have seen some variations, e.g., one full registration for the first accepted paper, one discounted "student" registration for subsequent accepted papers.

To further clarify, if you have two accepted articles which are co-authored by Author A and Author B, either only Author A or Author B register twice (and present both papers), or both Author A and Author B register once each and present one paper each. In the second case, it might be just a matter of speaking with the organizers to still let one author present both papers.