Let $X$ be a scheme and let $U \subseteq X$ be an open subset of $X$. If $U$ is an affine chart, then $\mathrm{Spec}(\mathcal{O}\_X(U)) = U$. Suppose now that $U$ is not an affine chart. By this, I mean that $U \ne \mathrm{Spec}(A)$ for any commutative ring $A$. Then the ring $\mathcal{O}\_X(U)$ is a commutative ring; what is the significance of $\mathrm{Spec}(\mathcal{O}\_X(U))$? Does it have any relationship to $X$? 


Feel free to add hypotheses to $X$ if it makes the answer more tractable or interesting. 



What is the significance of the Spec of non-affine structure sheaf sections?
Firstly, $\mathcal{O}\_X |\_U = \mathcal{O}\_U$ (restriction of sheaves), so you could just consider $U=X$. Then $\mathrm{Aff}(X) := \mathrm{Spec}\,\mathcal{O}\_X(X)$ is the universal affine scheme with a morphism from $X$, since we have an adjunction $\mathcal{O}^{op} \dashv \mathrm{Spec}$, i.e. $$\mathcal{S}ch(X, \mathrm{Spec}(A)) = \mathcal{R}ing^{op}(\mathcal{O}\_X(X), A)$$


$\mathrm{Aff}(X)$ is called affinization of $X$. I don't know of any deep meaning for affinization in classical algebraic geometry, since global sections of structure sheaves usually carry close to no information about the underlying scheme. For example, for any projective scheme $\mathcal{O}\_X(X) = \mathbb{k}$. However, affinization becomes much more important in derived algebraic geometry. I'm not going to make any intro into derived setting, but I'll quote one nice result.


In derived algebraic geometry the categories of quasicoherent sheaves on $X$ and on $\mathrm{Aff}(X)$ often happen to be equivalent. This is the case, for example, for $X$ which are finite CW-complexes, i.e. a finite homotopy colimit of a diagram of points $\mathrm{Spec}(\mathbb{k})$. For example, the affinization of circle $S^1$ is a spectrum of graded ring $$\mathbb{k}^{0,1} := \mathcal{O}\_{S^1}(S^1) = \mathbb{k}[x]/(x^2),\; \deg x = 1$$ This is the cohomology ring of $S^1$, and its spectrum is known in supergeometry as an "odd affine line" $\mathbb{A}^1[-1]$. This implies that for any affine scheme $\mathrm{Spec}(A)$ we have a series of equations for $B$-points ($B\in \mathcal{R}ing$)


$$\begin{eqnarray}
Hom( \mathrm{Spec}(B), \mathrm{Spec}(A)^{S^1}) & = & Hom( \mathrm{Spec}(B)\times \mathbb{A}^1[-1], \mathrm{Spec}(A)) \\
& = & Hom (A, B \otimes \mathbb{k}^{0,1}) \\
& = & Hom\_{graded} (A, B[x]/(x)^2)
\end{eqnarray}$$


The last set of graded ring morphisms is the same as $\mathrm{Der}(A,B)$ --- $B$-valued differentiations on $A$. The scheme that represents the functor of differentiations is the tangent bundle $\mathbb{T}\_A$. If we take grading into account, we get that $$\mathrm{Spec}(A)^{S^1} = \mathbb{T}\_A [-1]$$


The scheme $\mathrm{Spec}(A)^{S^1}$ is in fact affine. It is the (derived) spectrum of the Hochschild complex for $A$. Thus in derived geometry we automatically get the classical Hochschild--Kostant--Rosenberg theorem $$HC\_\*(A,A) = Sym(\Omega\_A[1])$$


Extra structures on $S^1$ and $\mathbb{k}^{0,1}$ give all structures on Hochschild complexes.


I learned this from the paper *Loop spaces and connections* of D. Ben-Zvi and D. Nadler, though the ideas are certainly much older. It also contains a nice intro to derived algebraic geometry.


