There are abandoned stations all over the NYC subway system, but I'm not too sure about how many segments of tunnel are abandoned.
However, the tunnel that Ray (Dan Akroyd) drops into with the slime flowing through it is supposed to be the Beech Pneumatic Subway, a subway line that was built in the 1870s that used air pressure to push a car down the track. A small demonstration track was constructed that ran from a station at the corner of Warren and Broadway and ran about the distance of a block. It lasted until the 1880s and was abandoned shortly thereafter. The station was destroyed in a fire and the tunnel was demolished when the current subway system was constructed (I think it was the IRT or BMT... can't recall off hand). So, that was a bit of Hollywood magic playing off a bit of urban legend (some say parts of the line still exist... it doesn't).
Here's more info. on Beech's subway...