This is really two questions. First, consider a normal toric variety $X\_\Sigma$. Its homogeneous coordinate ring 
$$R=\mathbb C[x\_1,...,x\_{|\Sigma(1)|}]$$ 
is graded by $A\_{n-1}(X)$. In analogy with projective space, I guess that there is an analogue of the Proj construction: homogeneous ideals of $R$ not contained in the Stanley-Reisner ideal $B(\Sigma)$ of the fan $\Sigma$. 



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> If $X\_\Sigma$ is projective, is $X\_\Sigma = Proj\_{B(\Sigma)}(R)$?
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Assuming this is true, I am curious about the case when $X\_\Sigma$ is quasi-projective; $R$ has non-trivial elements in "negative" degree. 



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> If $X\_\Sigma$ is quasi-projective, is there a analogue of Proj with $X\_\Sigma = QProj\_{B(\Sigma)}(R)$?
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Is a (quasi)projective toric variety (Q)Proj of its homogeneous coordinate ring?
No variant is necessary, $X\_{\Sigma}$ is $\mathrm{Proj}\_{B(\Sigma)} (R)$. Note: I'm assuming you already understand how this construction works in the projective case, so that I can jump in and start working an example.


Let's work through the example of $\mathbb{P}^2$ with a point deleted. The corresponding fan has three rays, in directions $e\_1= (1,0)$, $e\_2 = (0,1)$ and $e\_3 = (-1, -1)$ and with two dimensional faces $\mathrm{Span}(e\_1, e\_3)$ and $\mathrm{Span}(e\_2, e\_3)$. The ring $R$ is $k[x,y,z]$, with $x$, $y$ and $z$ corresponding to $e\_1$, $e\_2$ and $e\_3$. The grading is that $x$, $y$ and $z$ are in degree $1$. The ideal $B(\Sigma)$ is $\langle x, y \rangle$. 


So, the points of $k[x,y,z]$ correspond to those hemogenous primes of $k[x,y,z]$ which do not contain $\langle x,y \rangle$. Sure enough, that's $\mathbb{P}^2$ with a point removed! Note that we don't change the ring or the grading $R$, both of which are determined by the set of rays of the fan. We just change the irrelevant ideal $B(\Sigma)$.


See David Cox's paper [The Homogeneous Coordinate Ring of a Toric Variety](http://arxiv.org/abs/alg-geom/9210008) for the full generalities.


