Feedback resistors and op-amp instability
This particular op amp has a note in its datasheet on page 14 about the feedback resistor and how to choose components to ensure there is no oscillation:


  Feedback Components
  
  Because the input currents of the LT1213/LT1214
  are less than 200nA, it is possible to use high value feedback
  resistors to set the gain. However, care must be taken to insure [sic] that
  the pole that is formed by the feedback resistors and the input
  capacitance does not degrade the stability of the amplifier. For
  example, if a single supply, noninverting gain of two is set with two
  10k resistors, the LT1213/LT1214 will probably oscillate. This is
  because the amplifier goes open-loop at 6MHz (6dB of gain) and has 45°
  of phase margin. The feedback resistors and the 10pF input capacitance
  generate a pole at 3MHz that introduces 63° of phase shift at 6MHz!
  The solution is simple, lower the values of the resistors or add a
  feedback capacitor of 10pF or more.