Intro (verse) 0:00-0:05 A Major
Verse 1 0:05-0:27 A minor (with Picardy third)
Middle 8 #1 0:27-0:40* A (major or minor)
Verse 2 0:40-1:03 A minor (with Picardy third)
Middle 8 #2 1:03-1:21* A (major or minor)
Verse 3 1:21-1:46 A minor (with Picardy third)
Middle 8 #1 1:46-1:58 A (major or minor)
Verse 4 1:58-2:05* A minor (with Picardy third)
Coda (verse) 2:05-2:21 A (major or minor)
Comments: "I'll Be Back" marks the first Beatles recording in several categories:
First, there are two distinct Middle 8s, the second of which is slightly longer, but both of which end with the same music.
Second, the structure is perfectly palindromic (the same forwards as it is backwards), with the second Middle 8 serving as the centerpiece. The title even suggests the composer's knowledge of this fact on some level - "I'll Be Back" could refer to the structure of a rondo, where a particular theme or melody reappears several times. In this case, the song may be seen as a nine-part rondo (or a seven-part rondo with an intro and coda).
Third, there is an interesting tonal play between A major and A minor (which will appear in later songs, such as  "While My Guitar Gently Weeps). The intro is in A major, but the first verse is in A minor (but it ends in major with a Picardy third). All three Middle 8s explore other chords, but never to the extent of rivaling A as tonic. The coda, which fades out, alternates A major and A minor, offering no more weight either over the other. Although tonal ambiguity is not a trademark of Beatles music, the album A Hard Day's Night features two songs that are tonally ambiguous: "I'll Be Back" and  "And I Lover Her".
In addition, although less substantial and not unique, Verse 4 is abbreviated to prepare for the coda. Also, "I'll Be Back" features an interesting use of triplets - something I plan on blogging about in the future.