I'm currently listening to the Byrds' Turn! Turn! Turn! for the first time. I know some of these songs, but there are some I haven't heard before, including Gene Clark's "The World Turns All Around Her." In about the middle of the song, there's a drum part that's really close to the introductory drum part of the Beatles' "What You're Doing." I don't know drum terminology, but the drum part in "What You're Doing" sounds more drum-centric where the drum part in "The World Turns All Around Her" includes some of the other percussive elements of the drum kit (I think it's just that "The World Turns All Around Her" uses the snare drum, and "What You're Doing" doesn't). "What You're Doing" was on Beatles for Sale, which came out in 1964, and Turn! Turn! Turn! was released in late 1965, so that specific influence is possible, as far as chronology goes.
It's also pretty likely that the Beatles influenced the Byrds. In the liner notes for this album, Derek Taylor explains that George Harrison and Paul McCartney went to visit the Byrds while they were recording the album. And I remember reading somewhere that Roger McGuinn, Gene Clark, and David Crosby were doing Beatles covers even before the Byrds started. Plus McGuinn got his twelve-string Rickenbacker after seeing George Harrison's in A Hard Day's Night.