Mayhew Labs sells a Mux shield:
This shield can take up to 48 analog inputs which would support the large number of potentiometers the OP requested. They also provide some simple sample code for reading the mux's so you can process the signals. To get the data into the PC will require writing to the serial USB port and perhaps the use of the "Processing" software to handle whatever changes you would like to do to the data written.