First and foremost, as the title suggests, in what sort of order should I put author affiliations on the poster? Alphabetical? First author's affiliation, then alphabetical, or some other way (and do excuse me if a similar question has been asked and answered)?
Typically, affiliations are listed in the same order as order of authors. Listing them on a poster should be no different than listing them on a paper or in any other form of scientific communication.
Thus, for example, if you have the following set of authors:
Smith, affiliated with Ivy University
Chou, affiliated with Arrwun Tech
Patel, affiliated with Ivy University
Garcia, affiliated with McName Institute
Then you would typically list them as something like the following (adjusted for your desired format):
Smith, Chou, Patel, and Garcia
Smith and Patel are at Ivy University, Chou is at Arrwun Tech, Garcia is at McName Institute.
As for when authors have multiple affiliations: for any given author, affiliations should be listed in descending order of importance (as judged by the author). Typically there is a clear "primary" affiliation which comes first, and the order of any others after it matters much less. The order of the overall set across all authors is determined just the same way as when there are single affiliations, by listing all in order and combining duplicates.