From the library website:
The Huntsville-Madison County Public Library was founded in 1818, when Alabama was still a part of the Mississippi Territory, and it is the oldest continuing library in the state.
The library was first located in the office of attorney John Nelson Spotswood Jones, in the Boardman Building, which is now a part of Constitution Hall Park. The library also occupied space in the Green Academy from 1821 until Union soldiers burned the school during the Civil War. For several decades, the library moved to borrowed spaces multiple times, including the Murray and Smith Bookstore, the Y.M.C.A., the Gordon building, the City Fire Hall, and the Hundley building, until the Carnegie Library was opened on February 29, 1916.
From From "The Library World" column of April 1916, reprinted in The Library Journal:
Huntsville: The new Carnegie Library which cost $12,500 was opened to the public Feb. 29 with a book shower which was largely attended. The number of books contributed, with the previous donations, gave the library a fairly good start. Miss Carolyn Burke has been placed in charge as librarian. 
The operational history of the library stretches into the 19th century, and the library pictured — locally called "Fort Book" — was not the building opened in 1916. The 1916 date marks the opening of the Carnegie Library building (no longer in service), the opening of which catapulted the library into a major institution. So while February 29th does not commemorate either the opening of the library as a body nor the opening of the building that currently houses its volumes, it seemed the most significant date to commemorate in the library's history.