Are there any problems that can be caused by using resistors of large resistances (in the order of megaohms)?
I'm designing a feedback network that is just a voltage divider, and I want the feedback to drain as little current as possible from the circuit. The only thing that matters is the ratio between the resistors. So my question is: is there any reason why one would pick, for example, resistors of 1 and 10 Ohms instead of 1 and 10 MOhms?
There are many drawbacks to both low and high values alike.
The ideal values will fall in between very large and very small for most applications.
A larger resistor of same type will, for example, create more noise (by itself and through small induced noise currents) than a smaller one, though that may not always be important to you.
A smaller resistor will drain more current and create more losses, as you have surmised yourself.
A larger resistor will create a higher error with the same leakage current. If your feedback pin in the middle of your resistors leaks 1 μA when the resistor feeding that leak is 1 MOhm, that will translate to an error of 1V, while a 10k resistor will translate to an error of 10mV.
Of course, if the leakage is in the order of several nA or less, you might not care much about the error a 1 MOhm resistor creates. But you might, depending on what exactly you are designing.
Smaller resistors in feedback systems, e.g. with inverting amplifiers using op-amps, may cause errors on the incoming signal if the incoming signal is relatively weak.
It's all checks and balances, and if that's not enough information at this point, you might want to ask a more direct question about specifically what you are doing. With schematics and that.
In addition to the issues that @Asmyldof mentions, when using high resistances in the megaohms (and especially at 10M and more) environmental contamination such as dust, skin oils, soldering flux residue etc can easily reduce the effective resistance in unpredictable and time-varying ways.
In addition to other answers, also consider thermal noise. As your resistance goes up, so does the noise. If you want very accurate measurements, this may be an issue.
It's not unusual at all to use high resistances in dividers and feedback circuits for the reason you mention - to reduce current consumption and loading, especially for high impedance sensors, for example.
A few precautions should be taken to ensure predictable operation though. The board should be well cleaned before and after component placement to avoid contamination appearing as a parallel resistance. A good quality flux cleaner followed by an isopropyl alcohol swab is good for this.
If the circuit is to be operated in an unpredictable environment (like where there may be moisture buildup or high humidity) then a good conformal coating agent should be applied to the board and components, and baked out as per the manufacturers instructions to produce a sealed, high resistance moisture barrier.
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.
