Sorry but back-to-back zeners as described is a really nasty solution. Zeners get really "soggy" towards the knee and start conducting.
You should bias the zeners individually, giving them say 5mA current each. Then you can use this as a clamp point for a low leakage diode. We are talking about one zener, one bias resistor, and one low leakage clamp diode for the +5V clamp and this same circuit again for the -5V clamp. If you have +5V and -5V power rails you could of course just use schottky diodes to these power rails. You still need the series protection resistor from the opamp in all schemes.
schemes.
Note that putting zeners across the feedback network does not limit the amplifier output as desired. It drops the gain into the non-inverting input from 8.333 to 1, but that still means that increasing the non-inverting input will increase the output.
Whilst we are on the subject, your basic circuit bias is wrong. You have a non-inverting signal of +3.1V to +1.9V, that is nicely centred about the +2.5V reference level. When you put in +2.5V to the non-inverting input, given that the inverting input resistor is also at +2.5V, how much are you expecting at the output? If you think the answer is 0V then you flunked opamps 101.
Try again using nodal analysis, following the currents.