For many years it was widely held that this instrument was a bass harmonica, played by Charlie McCoy; and indeed Michael Krogsgaard's notes on the sessions say that McCoy played bass harmonica on the twenty partial or complete takes of the song. The notes for The Cutting Edge repeat this credit, so maybe it is written in one of the studio documents relating to the session. But the instrument we hear on this track just doesn't sound like a bass harmonica; it sounds rather like a harmonium (pump organ), but is more likely to be an electronic organ set to produce a similar sound - apparently Vox made an organ with this type of setting in the 1960s. Furthermore, Al Kooper has confirmed to me that he did play organ on this song, and that Charlie McCoy did not play bass harmonica, at least not when Kooper was around. He recalls that the organ track was either erased or just not chosen for the mix.
To hear what a bass harmonica played by Charlie McCoy really sounds like, listen to "The Boxer" on Simon & Garfunkel's Bridge Over Troubled Water album; or, if you happen to have the Collector's Edition of The Cutting Edge, listen to the last couple of seconds of "Stuck Inside Of Mobile With The Memphis Blues Again", Take 5, which sounds to me as though it has a bass harmonica doubling up with the bass guitar.