As you increase the temperature, the electrons, which are a cold Fermi gas, get more excited. Only a thin skin of electrons around the Fermi energy, of width kT, can do anything at all, like conduct heat. As the temperature increases, the electrons can conduct more heat.
The heat conductivity of electrons is greater than the heat conductivity of phonons, and together these account for the entire thermal conductivity. What is going on is that as the temperature goes up, the higher thermal conductivity electrons are carrying a larger fraction of the heat, and this makes the thermal conductivity go down.
In the absence of electronic thermal conductivity, for an insulating material, the thermal conductivity would go down with temperature, and this is also true for the thermal conductivity of just the phonons in the metal. But the electron contribution leads to this otherwise paradoxical effect.
It always helps to remember that a metal is never classical, the electrons are always quantum. A metal is a like a gigantic chemical bond involving all the atoms in a metal nonlocally, this is the conduction band, and the shared electrons have classically paradoxical properties. This is why the Drude model is wrong and the Fermi model is right.
I think that while Temperature is a major factor for resistance, the equation you wrote is not correct.
In a linear approximation,
$$R=R_0(1+\alpha(T-T_0))$$
where $R_0$ is the resistance at temperature $T_0$.
Exactly all factors that affect resistance, I think they are many, but these are the major ones (electrical applications).
Best Answer
For copper the temperature coefficient of resistivity is $3.9\times 10^{-3} \text{K}^{-1} $ and the temperature coefficient of thermal linear expansion is $1.6\times 10^{-4} \text{K}^{-1} $. They differ by a factor of about 24 so a change in temperature will cause a bigger change in resistance than in the linear dimensions of copper.
Resistance is given by $\frac{\rho L}{A}$ where $\rho$ is the resistivity, $L$ is the length and $A$ is the area of the copper.
Any thermal expansion will cause a bigger fraction change in the area $\propto \text{linear dimension}^2$ than in the length $\propto \text{linear dimension}$. So as a result of thermal expansion the resistance of the copper will decrease when the temperature.
So the net effect will be an increase in the resistance of a specimen of copper.
I have found that Kaye and Laby are an invaluable source of physical constants.