hse external oscilator on stm32f030
It will run without \$R_{ext}\$, but the clock will be noisier (because you don't have a low pass filter anymore), which may or may not effect the application. You may experience anything from more jitter on the clock which will affect timing or worst case an occasional upset.  

From the app note (app notes are good to read)


  The role of this resistor is to limit the drive level of the crystal.
  With CL2, it forms a low-pass filter that forces the oscillator to
  start at the fundamental frequency and not at overtones (prevents the
  oscillator from vibrating at 3, 5, 7 etc. times the fundamental
  frequency). If the power dissipated in the crystal is higher than the
  value specified by the crystal manufacturer, the external resistor
  RExt becomes mandatory to avoid overdriving the crystal.