Butler to Low tunnel
A potential Butler to Low tunnel is the Holy Grail of the tunnel system, but almost certainly just as mythical.
Thorough investigations of Butler Library have found that there are no tunnels going towards Low Library. This makes sense because Columbia's steam pipes run the periphery of the campus, underneath the buildings, not down the center where there are no buildings. Furthermore, grounds maps new and old document all the other tunnels, but show no existence of a Butler to Low tunnel.
Confusion with sewage system
Rumors about this tunnel stem from the existence of a bizarre square manhole cover on College Walk, but this manhole could just as easily lead to the NYC sewage system. 116th street between Amsterdam and Broadway was a city street open to traffic until Columbia obtained permission to close it off in the 1950's. If you're curious, though, feel free to pry open the manhole and tell us whether it's a tunnel or sewage that you see down there. I'd like to hear it. Many believe it could be the sewage system. An exact copy of the manhole can be found on 114th st.
A map from all the way back to the 1930's shows a 12" water main and a sewer line running across 116th street. It also shows the existence of numerous fire hydrants along what is now College Walk. This was even before South Hall (i.e. Butler Library) was built.
Speculation that tunnel is used for dignitaries
Some people speculate that the tunnel does exist, and that it is used to transport dignitaries, such as Koffi Annan, in private. However, the doors are alarmed, and disarming alarms is never easy. In April 2006, Pakistan's Prime Minister came to speak at the school. He entered from 114th street through the Carman delivery entrance and gave a speech at Low, but he was never above ground. This lends credibility to the rumor that there is some passage to Low from somewhere, but he most likely entered through the Mathematics tunnel system somehow.
Inside the Butler tunnel system is a Ventilation fan on the north wall. This fan does not show any evidence for a tunnel, but it is square and exactly the same size as the Mysterious Manhole on 116 st. Also, if one puts their hand to the entrance of this fan, one can feel cold air coming through, which is a surprise considering that the Butler tunnels are always hot due to the boilers. It is possible that this is the entrance to a very small (about 1 meter) tunnel connecting Butler and Low, but as of yet, there is no proof.
Mystery staircase in Low
Another intriguing piece of evidence is the existence of a mystery staircase in floor plans on the second to lowest floor of Low. The floors above and below it do not show it. It leads out of Low in the direction of Butler, but is under the level of the main steps. It immediately leads to the security break room, but no further. Blueprints of the original design verify this.
Evidence from maps
There are no known maps that show a Butler to Low tunnel. Furthermore, most tunnels were built when two buildings were constructed next to each other at the same time. To build this tunnel would have required considerable effort, when there are already two perfectly good north-south tunnels crossing College Walk.
A map from the 40's shows no tunnels out of Butler other than a future one to Carman and one John Jay.