4x4 Pressure Sensor Grid Pin Saving
Your schematic shows resistive sensors in parallel, so it will not work. You won't be able to distinguish the individual resistors when they are in parallel. This means some switching is needed.

http://sensl.com/downloads/ds/TN-Readout_Methods_for_Arrays_of_SiPM.pdf
http://www.sensl.com/downloads/irp/2013_Goertzen_Design_and_Performance_of_a_Resistor_Multiplexing_Readout_Circuit_for_a_SiPM_Detector.pdf

You will need switches... lots of switches...

Personally, I would try to keep things simple, and use two of these 1x16 analog muxes:

http://assets.nexperia.com/documents/data-sheet/74HC_HCT4067.pdf

The first one would take a current source as an input, and send it to one of the 16 resistors. The second one would take the voltage on the resistor which is being read, and forward it to the ADC.

This allows to ignore the switches' internal resistance. And since the same current source is used for all measurements, you can invest a little more into it to make it more accurate.