Often in power supply schematics they use two diodes at the inputs of opamps:
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What's the purpose of having two diodes at opamp inputs?
Electronic – Two diode at the inputs of opamps
circuit analysisoperational-amplifier
Related Solutions
The simplest way to eliminate your problem is to reduce the effective impedance of the source, and a voltage follower will do nicely
simulate this circuit – Schematic created using CircuitLab
The choice of buffer op amp is left to you, depending on the frequency response you need.
You can drive two amplifiers in parallel as long as you don't load the inputs too much (choose your resistor value in consideration of the drive strength of your signal sources).
You need a different summing amplifier circuit in order to get a non-inverting sum, however. Try a simple voltage divider connected between the two inputs, the result is the average of the input signals \$\frac{V_1 + V_2}{2}\$. Connect this to the positive terminal of an non-inverting amplifier with a gain of 2.
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The differencing circuit you have now looks ok.
Best Answer
It's because op-amps should have near 0 voltage difference between the inputs. The diodes are there so that a large voltage difference doesn't de-bias the input stage by causing the internal transistors to operate in non-linear modes. Sometimes those diodes are internal to the op-amp. Basically just input protection diodes.
That's one of the (many) reasons you don't want to use op-amps as comparators. Comparators will never have those diodes, while sometimes op-amp do have them, meaning that they will draw a lot of current and clamp when the input(s) swing.