No. take any homeomorphism that doesn't preserve the subgroup of $\pi_1$ that lift to closed paths in the covering. For example, take the 2:1 covering $S^1\to S^1$ take the product with the identity map on $S^1$. Let $h$ be the homeomorphism switching the factors.

In general, I believe an automorphism will lift if and only if the associated automorphism of $\pi_1$ send the subgroup of the covering to a conjugate.

Another way of saying this is that the category of coverings is equivalent to the category of $\pi_1$-sets, and a homeomorphism will lift if the corresponding twist of the $\pi_1$-set preserves its isomorphism class.

No. Take any homeomorphism that doesn't preserve the subgroup of $\pi_1$ that lift to closed paths in the covering. For example, take the 2:1 covering $S^1\to S^1$ take the product with the identity map on $S^1$. Let $h$ be the homeomorphism switching the factors.

In general, I believe a homeomorphism will lift if and only if the associated automorphism of $\pi_1$ send the subgroup of the covering to a conjugate.

Another way of saying this is that the category of coverings is equivalent to the category of $\pi_1$-sets, and a homeomorphism will lift if the corresponding twist of the $\pi_1$-set preserves its isomorphism class.