Can a postdoctoral fellow serve as a PhD committee advisor? Can a postdoctoral fellow, who holds a PhD in the relevant field, serve on a PhD student's dissertation committee? This might vary by field, but I'm curious whether there is any general trend on whether being a committee member depends on holding a PhD or holding a faculty position.
The main concerns for bringing in people to sit on a dissertation committee are satisfying the institutional regulations for committees (X members from outside the department, Y members from inside who aren't the advisor, Z members from outside the college) as well as bringing in possible expertise and exposure to additional ideas and approaches that may help the thesis candidate.

Normally a postdoctoral fellow will not count toward any of the composition requirements—one must be a faculty member or hold a similar position somewhere else (e.g., government research or industry). However, if one were to have a unique set of skills that someone thought it was essential to ask you to act as an additional member of the committee—this would typically be at another institution—there might be room and incentive to make something happen.