When the material is doped, the density of states in the conduction/valence band changes.
I'm not sure as to the actual physical mechanism involved in the lowering of the Fermi level, but when I think about it, I imagine there being more states, so it pulls the Fermi level towards that direction. If you dope Silicon with Boron, then you create holes, pulls the fermi level down and makes the material p type.
But the important thing is that the Fermi level is an imaginary 'filling' level that is very useful in understanding materials, but as far as i know, there is no compatible physical analogue, not like the conduction band and valence band which have actual physical meaning. (Correct me if I am wrong here)
Finally, I don't know what you mean by Fermi level changing with temperature. The band gap changes with temperature according to the Varshni equation, and the accurate position of the Fermi level in an intrinsic semiconductor material is slightly affected by the temperature, but not by much and generally negligible when solving questions.
If you are referring to doped materials, then I would argue that any temperature variations of the Fermi level will be dominated by the dopant effect.