First lets consider the problems using LOW resistor values, with opamps.
The biggest problem is the opamp's limited output current. Often 20 mA is the maximum for accurate performance. Yet, 1 ohm and 1 volt require 1 ampere. It is not available. Thus you must design with higher values.
Another problem with LOW values is the thermal distortion, as self-heating causes big temperature changes and big resistance changes. Using 1 ohms and 9 ohms, to set gain in feedback loop of opamp, causes the 9 ohms to dissipate 9X the power. At 1 millivolt input, the 1mA current may or may not cause detectable distortion. Walt Jung discussed this, for Audio Power Amplifier feedback dividers.
Now for HIGH value resistances:
A problem with higher values comes with the capacitance on the -VIN pin of the opamp. The phase shifts ---- 1 megohm and 10 pF have a Tau of 10 µS, thus a 45 degree phase shift at 16 kHz ---- it leads to peaking, instability, and oscillation. The cure is to use tiny capacitors in parallel with the high value Rfeedback resistors...another component to buy and install.
High resistances leave the circuit vulnerable to Efield interferers. The capacitively injected charges will find a return path. A 10Meg Ohm resistor, facing 160volt 60Hz wiring at 4", coupling into 14mm by 1mm PCB trace, induces 1.5 millivolts of 60Hz. At the 1Kohm level, the interference is 10,000X smaller.
Lets also examine an LDO, providing regulated 2.5 volt output for any Vunreg over 2.7 volts, with standby current < 1uA per the datasheet. What do we know about the output noise of that LDO?
simulate this circuit – Schematic created using CircuitLab
We know this LDO has at least 60 microvolts RMS output noise, because of the 12Million Ohms (times 2) feedback resistors. At least 60uV, because the internal opamp has high noise (at very low currents, expect high noise) and the 1.22 volt BandGap has high value resistors.
I recall an LDO, with 1uA Iddq, showing poor PSRR above 100Hz. Turns out the Vin metallization was above the 12Meg Ohm voltage dividers. Any trash coming into the LDO was directly injected into the servo-amplifier loop. Learn to visualize these problems. The original designer stated "the parasitic extraction did not show this as a problem." Learn to visualize these problems.