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 do say that McCoy played bass harmonica on the twenty partial or complete takes of the song. However, it certainly doesn't sound like a bass harmonica*; it sounds more like a harmonium (pump organ), or maybe an electric 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.
However, the myth of the bass harmonica lives on, and was recently given an airing by Sean Wilentz in his article for Oxford American on the making of Blonde On Blonde.
*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.