Are there copyright laws that protect public librarians from having their lectures filmed and posted?
There is a link below from Baylor University that summarizes Fair Use.
Essentially, the answer is, "it depends". In some contexts, lectures can be recorded, such as educational and journalism purposes.
However, your library (or city department) may have policies that govern filming or recording in the library.
Ultimately, if there is a nuanced copyright or privacy issue, a lawyer may need to be consulted if an agreement cannot be reached with the person recording the lecture.