Why use an input resistor in this current sense circuit?
The 200 Ohm resistor on the inverting input has no effect for an ideal op amp, which has no input current.

But real op-amps require a (very small) bias current, and also have offset currents flowing from the non-inverting input to the inverting input.

Because of the offset current it's best practice in a precision circuit to have equal impedances feeding the two inputs of the op-amp.

In the case of the LT1637 with 5 V supply, the offset current could be as high as 15 nA. If the input impedances weren't balanced this could cause an error of up to 3 uV, corresponding to an error in the current measurement of 15 uA.


  is there any downside to sizing the 200 ohm resistors as e.g. 10k?


There's no real issue with a small change in that resistor value (for example to 211 Ohms or something), but no advantage either. 

If you were to increase the R1 resistance to 10 kOhms, I'd start to worry about Johnson noise generated by the resistor. But I haven't looked at the effects carefully, and of course the maximum acceptable noise depends on your system-level requirements.