Non-Inverting Amplifier
One of the reasons for adding a resistor at the input of a non-inverting amplifier is to reduce the voltage offset in high-precision measurements, which is caused by its finite input resistances as described by Maxim and TI.
Essentially a op-amp has a very large, but finite input resistance (e.g. LM358 has a common-mode impedance of \$4M\Omega||1.5pF\$), and consequently some current flows into the inputs.
It is described more in detail in the Maxim application note, but here is what might happen if the input resistor is not used:

Consider that no resistor is used and that \$V^+\$ is tied to ground. By doing so, the input resistance of \$V^+\$ if being shorted to ground, thus not playing any role anymore.
Disregarding that the input/output common mode voltage for the moment, the feedback would force the inverting input \$V^-\$ to match the \$V^+\$ voltage.
Since the voltage across \$R_{P1}\$ in this case would be zero, no current flows through it.
Replacing now the input resistance with a current source, and solving for the output voltage, yields

$$V_{out}=R_{S1}\cdot I_{R,V_{IN^+}}$$
This means that even if the non-inverting input was to be shorted, some output voltage would appear. This problem can be mitigated by adding a resistor in the non-inverting input with a value approximated to:
$$R_{V_{IN^+}}= R_{S1} || R_{P1}$$
The entire derivation can be found in the links provided.