I learned enough of each part to be able to compare them. Almost all of the bass part during the verses of "Go Go Go" alternates between these two measures:
There are two similarities between these two parts. First, both phrases have triplets on the third and fourth beats of the measure (although this is in only every other measure in "Go Go Go"). Second, both phrase arpeggiate chords in the same way. They're different chords, but the bass plays the root, third, fifth, and sixth. The bass part in "Go Go Go" continues ascending to play the root an octave higher in the second measure, and the bass part in "Old Brown Shoe" plays a second when it descends, but they start out the same (although, admittedly, a root-third-fifth-sixth arpeggiation in the bass isn't that distinctive).
After I realized this similarity, I thought about the bass part in "I Saw Her Standing There," which, adjusted for key, consists of the same phrases as the bass part in Chuck Berry's "I'm Talking about You." Initially, I thought Paul McCartney nicked an-other bass part from a Chuck Berry song, but then I lookt in the liner notes of Anthology 3 to find that all of the parts in the demo of "Old Brown Shoe" (so, obviously, including this phrase) were played by George Harrison. Evidently, it was he who came up with this phrase. I'm not sure whether the bass part in Berry's "Go Go Go" was any influence, but the two parts do have a certain similarity and the Beatles were obviously familiar with other Chuck Berry songs.