Non-Inverting Op-Amp noise on the output when touched
As with all noise problems your BEST solution is to prevent the noise from  getting picked up by the circuit in the first place. As such you already have a solution, which is to electrically isolate the sensor from prying fingers. 

As the doctor said when I told him.. "It hurts when I do this!"

"Well then, don't do that!"

C1 in this circuit turns it into a low pass filter. The feedback impedance will be significantly reduced for higher frequencies on the signal. As such, the larger the capacitance the less "noise" will make it through. At very high frequencies \$R_{FEEDBACK}\$ --> 0.

However, for a circuit like this, finger interference means you will be injecting a significant amount of low frequency, mains hum, noise. As such, the capacitor would need to be quite large to remove it.

Since you are planning on multiplexing the signal from various sensors, the effect of a large capacitance here is problematic for settling time delay as you have intimated.

As such, and if it is not possible to properly insulate and isolate the sensor from fingers and local RF noise, this is really not a good circuit to use for this application. 

Moreover, using the GPIO pins directly to switch between sensors also adds an issue of inter-channel interference and a general inability to REALLY get the bottom end of the sensor hard to ground. Further, switching like that creates what can be problematic transient response issues in the op-amp.

It would be better to receive each sensor value individually, with appropriate noise filtering, and then select the appropriate output to feed to wherever the signal is intended to go.