It seems as if there is a 220K pull-down resistor on analog pin 0 So I am building a project that uses the analog pins for precise measurements of resistance through a voltage divider and I have been having some seemingly unsolvable problems with nonlinear response to linearly changing resistances when using the A0 pin to measure. 

I first realized this was the problem when I had a voltage divider consisting of a 76.8K and 100K 1% resistors with Vout connected to A0, and I was consistently getting wrong values. I measured the output voltage of the voltage divider and got 1.7V, instead of the ~2.7 volts I should be getting with my supply. When I disconnected the wire to A0, the voltage rose to the expected level. 

I disconnected power and measured resistance between the analog pins of the Atmega328 I was using and ground and found that A0 had 220K of resistance, while all other pins measured as infinite. When using the other pins, the voltage divider returned the expected values and was able to measure a wide range of resistors with values just as accurate as my multimeter. I know I have changed a lot of low-level things with code (analog reference, the comparator, ADC clock prescalers, using analog pins as outputs, etc.) but I don't think any of this could persist while the chip is not powered on. Does doing any of these things connect a resistor in the way I was describing, and is there anything that could cause this to stay "stuck" this way?
They don't connect a resistor as such, but analog input pins do have an impedance which can affect your readings.

If your output impedance (the resistance of your resistors) is within about a tenth of the impedance of the input pin then you will (and have) noticed skewing of the results.  This is normal.  The input impedance is somewhere in the region of 1MΩ - 10MΩ.

You have to keep your output impedance to a value below the order of a few tens of kΩ to minimise the effect the measuring of the value has on the value.

If you want to measure higher resistances then you will need to buffer the voltage with a very high impedance op-amp buffer.

Incidentally, your volt meter probably has an input impedance in the region of 10GΩ - around 1000x higher than the Arduino's ADC.