Hi,
If you connect a low value resistor, then gain will increase. The Opamp won't get hurt.
The disadvantage is: lower frequency response, increased distortions, increased gain error.
But it will have less output noise than the two opamp solution and it will have less output offset than the two opamp solution (if both curcuits have the same overall signal gain). Almost independent of the second amplifier.
It depends on your specifications:
Signal frequency, waveform, source impedance, offset voltage, offset voltage drift, distortion, gain, gain error, phase shift, noise...
Klaus