Which of the following quantities does not change when a resistor...

When a resistor is heated by the current, its temperature increases.

The number of free electrons in a metallic resistor depends on the nature of the material and remains practically constant with changes in temperature.

The relaxation time $\tau$ decreases with an increase in temperature due to more frequent collisions of electrons with the vibrating lattice ions.

Resistivity $\rho = \dfrac{m}{n e^2 \tau}$ increases because $\tau$ decreases.

Resistance $R = \rho \dfrac{L}{A}$ increases as resistivity increases.

Drift speed $v_d = \dfrac{e V \tau}{mL}$ decreases because the relaxation time $\tau$ decreases (assuming the potential difference $V$ from the battery remains constant).

Therefore, only the number of free electrons remains unchanged.