I know it has small phase margin.. but why is it oscillating not at rails?
The designer might have intended to make a band pass or low pass filter, but neglected to put any impedance/resistance between the capacitor and the op amp. But this is purely speculation and EE.SE has no place for that. 

One thing to remember is amplifiers and ADC's have input impedance, in this case R1 is 30kΩ (from the datasheet)


I can't see a reason to keep the capacitor on the board in its current configuration it's loading the op-amp with capacitance, either convert the circuit into a low pass filter (and cut back on your noise) A good reference is Op Amps for Everyone from TI or any op amp design guide from an analog manufacturer. 

Learn how to do filter calculations, check the LT1001 datasheet for info on capacitance (fig 1001 G20) for info on overshoot. Design stuff with hand calculations, then maybe do a simulation in LT spice and then implement the changes and test them to see if they improve your situation.