The DOL explains:
The FLSA recognizes the generosity and public benefits of volunteering and allows individuals to freely volunteer in many circumstances for charitable and public purposes. Individuals may volunteer time to … non-profit organizations as a public service and not be covered by the FLSA. Individuals generally may not, however, volunteer in commercial activities run by a non-profit organization.… A volunteer generally will not be considered an employee for FLSA purposes if the individual volunteers freely for public service, religious or humanitarian objectives, and without contemplation or receipt of compensation. Typically, such volunteers serve on a part-time basis and do not displace regular employed workers or perform work that would otherwise be performed by regular employees.
Thus, as to the above job posting, the analysis hinges on whether the sought Assistant Adjunct Professor is related to UCLA's charitable activities or its commercial activities. Since UCLA charges tuition to students to attend classes, including the chemistry class(es) this to-be-hired adjunct professor will teach, it's difficult to construct an argument that the position supports the university's charitable purpose. Indeed, it directly supports the school's commercial activities.
If, for example, the university was filling a position for someone to go into the community to offer educational services to at-risk youth, I could envision an argument for charity over commerce. But this is not that position. UCLA is hiring a professor to teach classes on campus to enrolled, tuition-paying students. This position certainly appears to be purely commercial in nature, and thus must be compensated. If UCLA fills this position as advertised without compensation it's in for a world of legal hurt.
I'm surprised that UCLA's legal department would allow this job posting (assuming it knew about it). I'm very curious to see if it changes now that people are talking about it and calling the university out for it.