Is the mapping from a scheme to its global sections a closed map? This is a question posed to me in private communication by this user.
Given a scheme $T$, let $\Gamma (T) = Mor (T, \mathbb{A}^1)$ be the ring
of global sections. Note that there is a canonical map
$\phi : T \rightarrow Spec (\Gamma (T))$.
Is $\phi$ a closed mapping onto the image, ie. is $im\ (Z)$ a
closed subset of $im(T)$ for all closed subsets $Z$ of $T$ ?.
 A: Here is a ''natural'' example as expected by Martin. Let $T$ be the projective line over ${\mathbb Z}$, minus a rational point $x_0$ of the closed fiber at some prime $p$. Then $O(T)=\mathbb Z$ (direct computation or use Zariski's extension theorem for normal schemes), $\phi$ is just the structural morphism and is onto. Let $Z'$ be a section of the projective line passing through $x_0$. Then $Z=Z'\cap T$ is closed in $T$, but $\phi(Z)$ is not closed in $\phi(T)$ because it is the image of 
the composition $Z\to {\rm Spec} O(Z)\to {\rm Spec} O(T)$, and as $Z$ is affine, the image is just the complementary of the closed point $p$. 
A: Note that if $T$ is reduced, this morphism has dense image.
If $T$ is a locally compact totally disconnected hausdorff space, it can be given explicitely a reduced scheme structure, which is affine iff $T$ is compact. Besides $T \to Spec \Gamma(T)$ is an dense open immersion. Thus it is not closed as long $T$ is not compact. I've proven this here. Thus a counterexample would be $T = \mathbb{Q}_p$.
I expect that others will show you more "natural" examples.
A: [Added: I misread the question, and in fact this answer does not answer the OP's question,
but rather the following question: is $\phi(T)$ closed in Spec $\Gamma(T)$, which is
a different question.  Probably the upvotes can be attributed to the link to the stacks project!]
If $T$ is quasi-affine (i.e. admits an open immersion into affine space),
then the map $\phi$ is an open immersion, and in fact Spec $\Gamma(T)$
is the initial object in the category of affine schemes containing $T$
as an open subscheme.  
In particular, in this case $\phi$ has closed image
if and only if and only if $T$ is in fact affine.  [Added: As Qing Liu points out in a comment below, in this quasi-affine situation, $\phi$ is in fact a closed map onto its image.]
Thus if we  take $T$ to be ${\mathbb A}^2_k \setminus \{0\}$ for some
field $k$, i.e. affine $2$-space with the origin removed,
then we get an example of $T$ where this map is open with non-closed
image (since this $T$ is quasi-affine but not affine).  Note that
Spec $\Gamma(T) = {\mathbb A^2}_k$.
(This is a geometric analogue of Qing Lui's more arithmetic example;
what both have in common is that a closed point was removed from a 2-dimensional
affine scheme, so as to make a quasi-affine scheme that is not affine.[Added: I also misread Qing Liu's example; my remark would apply to the affine line over ${\mathbb Z}$ with a closed point removed; Qing's example is more complicated, since it is actually dealing with the OP's question.  One can make a geometric analogue of Qing's example by deleting a closed point from ${\mathbb A}^1\times {\mathbb P}^1$; more geometrically still, remove one of the lines of a ruling from a projective quadric surface, and then remove an additional point.])
EDIT: In the definition of quasi-affine, one should also require that $T$
be quasi-compact.  (The stacks project
is a terrific resource for these foundational definitions in scheme theory,
particularly with regard to finiteness and separation issues.)
Note that if $T$ is any quasi-compact scheme, then the map $T \to$ Spec $\Gamma(T)$ has dense image.  (If $f \in \Gamma(T)$ and $D(f)$ is the usual affine open in Spec $\Gamma(T)$,
i.e. Spec $\Gamma(T)_f$, then if $\phi^{-1}(D(f))$ is empty, it must be
that $f$ is locally nilpotent on $T$.  Since $T$ is quasi-compact this implies
that $f$ is actually nilpotent, and hence that $D(f)$ is empty.)  As Martin notes
in his answer, this is similarly true if $T$ is reduced.
It need not be true if $T$ is non-reduced and non-quasi-compact (since $T$
may then admit locally nilpotent sections of $\mathcal O_T$ that are not
globally nilpotent, e.g. $T = \coprod_n$ Spec $k[x]/(x^n)$).
