Every noncompact manifold admits nonzero vector fields, or more generally,
vector fields with any specified set of isolated zeros along with the behavior near
that zero.  

 However, if you have information of the behavior of a vector field
near infinity, or just in a neighborhood of the boundary of a compact set, there
is an index theorem.  Perhaps this is the case with your $\mathbb T^! \times \mathbb R$.

One formulation is this:  let $N^n$ be manifold, and let
$M, \partial M \subset N$ be a compact submanifold. Let $X$ be a vector field that is
is nonvanishing in a neighborhood of $\partial M$.  Suppose that the set of tangencies
of $X$ to $\partial M$ consists of a codimension one submanifold $T$ of $\partial M$,
 so we can write
$ \partial M$ as the union of two submanifolds with boundary, $\partial M = B_+ \cup B_-$
where the flow of $X$ is outward on $B_+$ and inward on $B_-$.
Then the Poincaré-Hopf index for $X$ in $M$ is $\chi(M) - \chi(B_+)$

Here's one proof:   triangulate a neighborhood of $N$ so that $\partial M$ and $T$
are subcomplexes, and so that $X$ is *transverse* to the triangulation except
near the singularities, in the sense that in any simplex, the foliation defined by
$X$ is topologically equivalent to the kernel of a linear map in
general position of the simplex to $\mathbb R^{n-1}$.  Put a $+1$ at the barycenter 
of each triangle of even dimension, and a $-1$ at the barycenter of each triangle
of odd dimension.  Think of $X$ as a wind that blows these numbers along, so
that after an instant, all numbers (except for exceptions near the zeros of $X$)
are inside an $n$-simplex.  In any typical simplex, all the signs cancel out.
However, the numbers on $B_+$ (which includes $T$) are blown away and lost. Near any zero of $X$ there is a mismatch,
which can only depend on the local behavior of $X$.
From the special
case that $N$ is a closed manifold and $M = N$, it follows
that is the Poincaré-Hopf index.