Suppose that $p: X \to Y$ is the universal covering of some connected and locally path connected space $Y$, and that $\phi$ is a deck transformation. Is $\phi$ homotopic to the identity on $X$? If so, why?
I'm asking because this fact is used in this question and I'm not sure why it should be true.
Not necessarily. The antipodal map is a deck transformation of $S^n \to \Bbb R P^n$, but it's not homotopic to the identity for even $n$.
Still, the claim is true in the question you link to. The lifts in the question fix base points. Under the given conditions, if two lifts agree on one point, they agree everywhere. See Hatcher's Algebraic Topology, propositions 1.34 and 1.37.