What happens to the output (S) in an NTE74C4067 multiplexor if multiple channels are active?
S0..S3 are for selecting the active channel. They are digital inputs, not outputs. 

Only one channel at a time (at most) is connected to the "common input/output". All the others are floating. If /E is '1' then the common input/output is floating (no channels are connected). 

It acts just like a selector switch- with an 'off' position. When the selection inputs are 0x0101 and /E is '0' then channel 5 is active, which means that I5 (pin 5) is connected to common input/output (pin 1) through something like 100 ohms typically. The other I0..I4 and I6..I15 are floating. 

The multiplexer acts like a resistor when it's 'on', so it doesn't really care if you use the I pins as outputs or as inputs. It also doesn't care if the signals on I0..I15 or common input/output are logic levels or are some other voltage between the power supply rails- 100mV or whatever. Hence the name Analog Multiplexer/Demultiplexer

If you want to connect multiple things at once, you can use individual switches, or if you have a many-to-many matrix, a crosspoint switch.