This is one of those tunes tunes where I think analyzing the three chords in terms of classical tonal harmony is a somewhat mistaken approach. I'll explain why:
In terms of roman numeral analysis, you see i7, bVI7#11, iv7, i7.
The movement from i7 to a bVI isn't so uncommon, the dominant flat six sound also being associated with Bernies tune and functionally being a tritone sub into the V chord. However, first problem is we usually associate tritone subs with alt scales. Second problem is that we don't follow up with the V chord, instead going to vi chord which is a sub-dominant function.
So, instead of getting yourself messed up in that kind of functional garbage (I know it's not GARBAGE per se, but bear in mind that's not useful towards playing and improvising on this tune and I doubt Herbie was even thinking of this when he wrote the tune), let's instead think of each different chord as a separate tonal center, denoted by a chord-scale relationship. So we simply start on F Dorian, go to Db Lydian Dominant, then to D Dorian, and then back to F Dorian. Usually when tunes like this are composed, the composer isn't necessarily thinking functionally as in classical music, but instead looking for interesting shifts in "color."
Or to say it all more simply: This tune is modal, as in hgiles' reply.