Who should I ask for reference letters for academic positions?
Short answer: Extra letters do not improve an application, but could undermine it.

Elaboration: One advice often given in career center presentations or online articles about post-doc/TT applications is that the application packet should only contain the documents (and the number of documents) requested. If the announcement asks for 3 letter of reference, it means the hiring committee expects 3 letters, not 2 or 4. This is a simple yet often overlooked fundamental criterion of a successful application.

Assuming you are planning to supply only the requested number of letters and the issue is which letters to include, I recommend sticking with the traditional approach of letters from the individuals who you have worked with in subordinate capacity with during your PhD studies. This typically includes your dissertation advisor/committee chair, perhaps another committee member, or (if different) a PI on a grant you worked on, whether in or outside of your department (e.g. an affiliated research center). 

I would advise against letters from very junior faculty at other institutions (e.g. your recent peers) as they carry relatively little clout and the hiring committee might get the wrong idea if they suspect your choice of using them as reference may have been forced, to some extent, by circumstances that prevent letters from more reputable/senior colleagues (in other words they may come across as less convincing and perhaps even suspicion-provoking reference choices - and you don't need that).

Generally, I would advice including letters from the individuals (faculty) you have worked with most recently. The dissertation advisor is an unavoidable choice and a must. Beyond that, if you worked with other faculty who were PIs or partners you have collaborated/co-authored with, choose the individuals you have worked with on most recent projects/publications as consistent with the chronology of your academic appointments/experience in the CV.

It is also a well known and accepted (if not publicly advertised) practice to pre-write letters of recommendation to save your references' time. Whether you have done this already or not, good to keep in mind. Just ask your reference in a matter-of-fact way if they prefer to author the letter or could use a summary draft (or at the very least, a current copy of your CV).

Good luck with your apps! Let us know how it goes!