Given a self-map $h$ of a (closed?) manifold, is there a vector field $\xi$ with flow $\Phi_t$ such that $h = \Phi_1$?

If $h = \Phi_1$ for some flow $\Phi$ then clearly $h$ is homotopic to the identity. So, one can obtain many counterexamples by simply constructing self-diffeomorphisms that are not homotopic to the identity, and one can detect this using reasonably simple invariants of algebraic topology.

For one example, any orientation reversing diffeomorphism of a closed, oriented manifold $n$-manifold $M$ is not homotopic to the identity, because it induces the "multiplication by $-1$" map on $H_n(M;\mathbb R) \approx \mathbb R$. (This works as well for connected oriented manifolds, but the obstruction is a bit harder to describe).

For another example, the torus $T^2 = S^1 \times S^1$ has fundamental group isomorphic to $\mathbb Z \times \mathbb Z$. For any matrix $M \in SL(2,\mathbb Z) = \text{Aut}(\mathbb Z \times \mathbb Z)$ there exists a diffeomorphism $\phi_M : T^2 \to T^2$ such that the automorphism of $\pi_1(T^2) \approx \mathbb Z \times \mathbb Z$ induced by $\phi_M$ is given by the matrix $M$. Thus $\phi_M$ is not homotopic to the identity if $M$ is not the identity matrix.