Why must resistance never be measured in an active circuit?

First of all, in an active circuit, there are voltages flowing through various nodes. If you try to measure the resistor, those voltages will cause error in your meter, or may damage your meter. Second of all, even if you cut the power, any loop in the circuit around the resistor will affect the measured impedance. Unless you are certain that only the resistor will see the current introduced by the meter, you must take the resistor out of circuit to measure it.