Constant voltage source for unkown load
If you're trying to use the test circuit from the datasheet then you've got an error in copying the circuit.

Figure 1. From the datasheet.

Figure 2. What you created.
You have your load connected between U1 output and U2A non-inverting input. The input will have a very high impedance and the datasheet says the input bias current is 50 nA. You can't sink current into the non-inverting input.

Update after edit:

Figure 3. Modified schematic.

I am designing a circuit for maintaining constant voltage for unknown load ...

No. This circuit will control the current, not the voltage.

I had done some simulations on Tina before making the PCB with the components. The simulation worked as expected.

What did you expect? (You haven't told us.)

What you should expect is that when stable the negative feedback should bring U1 inverting input to virtual ground. The voltage there should be really close to 0 V.
Since R1 and R2 are both 10k that means that the output of U2A will be -1 × Vin.
Since U2A is configured as non-inverting buffer that means that Vout will be -1 × Vin too.
This means that U1 will have to swing negative for a positive input signal.


Now I have assembled the PCB. For testing purpose I have shorted the 100Ohm Resistor and input of LMP7702. When I apply for example 1V at the Input_Signal, Vout comes out ~536mV and output of LMP7702 comes at around ~884mV. This should obviously not happen as the LMP7702 is acting as buffer.

Try this:

Leave R3 in circuit and short out R4.
Set R5 = 1 kΩ. It will make the maths easy.
Set the input to 2 V.
If everything is working Vout should go to -2 V. You'll have 2 mA going through R5.
U1 output should be \$ \frac {100 + 1000} {1000} V_{out} = \frac {11}{10} V_{out} \$ (-2.2 V).

Update your question with your results and we'll go from there.