In Chapter 1, section 5.4 of Arnold's ODE book he is talking about diffeomorphisms on direction fields. To prove that integral curves of the field in $M$ are mapped to integral curves of the image of this field under diffeomorphism $g$ in $N$, the vector field needs to be reconstructed from the direction field.
He does it a paragraph earlier: for every $y$ in $N$, find $x=g^{-1}y$ in $M$. The direction field at $x$ is a line in $T_xM$. Take a vector $v\neq0$ along the line and get $w=g_{*x}v\neq0$ (as far as I understand, since $g$ is diffeomorphism, $w=0$ would violate smoothness of the reverse mapping $g^{-1}$) in $T_{g(x)}N$. Vector $w$ is independent of the choice of $v$ due to linearity of $g_{*x}v$ and will define the line of the image of the direction field.
So far, he hasn't been talking about the field being smooth. Now, in Problem 1 it is asked if a smooth vector field can be reconstructed from a smooth direction field and the answer says "No, if the region (I guess in a more general case, manifold) $M$ is not simply connected".
So, my first question is if I understand it correctly that crucial part here is about the field being smooth?
Secondly, I cannot see how the smoothness would be violated if the region $M$ has a hole in it as in Fig. 58, illustrating the answer.
PS. I am quoting from Russian edition of the book.
Consider, on a Mobius band, the line field that runs orthogonal to the central circle. You can pick a nonzero vector in at $\theta = 0$, but by the time you've come back around to $\theta = 2\pi$, the nonzero vector is pointing in the other direction, and there's no way to adjust it to make a smooth nonzero field.
I guess the answer to the question you actually asked is "no, the key thing is non-zero-ness," and for the second, it's not the smoothness that's violated, but the non-zero-ness.
Given your math.se login name (mobiuseng), there's a small irony in this answer. :)