FILM FUN ran from July 1915 to September 1942. Actually, however, it traced its roots back to the late 19th century, having begun as JUDGE'S SERIALS (1887 to January 1890), reprinting material from the humor magazine JUDGE. This became JUDGE'S LIBRARY (February 1890 to July 1912) and then THE MAGAZINE OF FUN (August 1912 to June 1915). (Source: Richard E. Clear, Old Magazines.) Apparently FILM FUN did not continue on after 1942 under some other name.
OK, going by the covers it's hard to take FILM FUN seriously as a movie magazine. Well, it didn't take itself seriously. In the beginning it covered humorous movies and actually had comedians on the covers. By 1921 it was running painted covers by the likes of John Held, Jr. (his usual topics), and by 1922 Enoch Bolles was painting for it. Bolles did women in bathing suits and women in skimpy clothes; some believe they were called "pin-ups" (how quaint that sounds today!). Meanwhile the magazine continued with stories about movies, biographies of the stars, and lots of pictures. Maybe they were racy by the standards of the day, but today we find them, well, quaint. From mid-1923, until just before it folded in 1942 (after changing to photographic covers, partly because it was under threat of an obscenity prosecution), Bolles painted almost every cover for this magazine.
If you like the later covers, you should definitely check out Jack Raglin's blog about Enoch Bolles.