At Brandeis University, where I went to college, we heard the name Irving Fine pretty regularly, because he'd been an important early member of the music department and because of the annual concert in his memory. I therefore have absolutely no idea of how well-known he was and is elsewhere in the musical world, although in ten years of peering at offerings from local music organizations, I can't remember seeing anything of his on a program. Certainly his marvelous Alice in Wonderland songs live on, but the rest?
It seems there is a revival of interest. Recently, the NY Times published a terrific Will Robin* article on Fine. (It also discusses Harold Shapero and Arthur Burger, fellow members of the Brandeis music department, both of whom I took classes from.) Ethan Iverson has a long discussion of Fine and his music at Do the Math. Read 'em both, and then pick up some recordings of Fine's music.
Update: Will Robin himself tells me that the Boston Modern Orchestra Project is recording all of Fine's orchestral works. The CD can be pre-ordered and will be published in January, 2015.
* I interrupt myself to report that search at the NY Times is so fucked up that a search for William Robin returns Robin Williams first. NO. But that's what I get for not putting his name in quotation marks.