generalized principal open set Let $V$ an affine variety. A principal open set is a set of the form $V(f) = V \setminus\{f=0\} $. A well known theorem states that all such sets are affine varieties, and moreover (Shafarevich, p.50) have coordinate ring $k[V(f)]=k[V][f^{-1}]$.
Now, I am interested in a more general situation - consider again an affine variety $V$, but now look at
$$V_{f_1,\dots,f_t}=V\setminus \{f_1 = \dots = f_t =0\}$$
These are quasiprojective varieties, since $V_{f_1,\dots,f_t}=\bigcup_{i=1}^{t}V_{f_i}$.


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*How can it be shown that such a set is not an affine variety (if it is indeed the case)? In general, what tools are used to show that a given quasiprojective variety is not affine?

*Are such sets projective varieties? If so, they are automatically not affine, since the only variety which is affine and projective is one point.

*What about the ring of regular functions of $V_{f_1,\dots,f_t}$? Does it equal to $K[V][f_1^{-1}]\dots[f_t^{-1}]$? If not, what can be said about it?

*After finding it, can this ring be used to show that the variety is not affine (the equations may be very unpleasant)?

 A: 1)  Your varieties $V_{f_1,\dots,f_t}$ are almost never affine, except in very degenerate cases (like for instance $f_1=\dots=f_t$ !).
The basic tool is the following very general and powerful theorem:  
Given an arbitrary subvariety $Y\subset X$ of the arbitrary variety $X$, if the complement $X\setminus Y$ is affine, then $\operatorname {codim} _X(Y)=1$ 
Notice that we don't assume $X$ affine here.
This is a scandalously underappreciated result due to Goodman (Ann. of Math,  vol.89, page 162), which doesn't seem to be mentioned in the standard books .
The codimension $1$ condition is of course not sufficient as witnessed by the example $Y=\{*\} \times \mathbb P^1\subset X=\mathbb P^1 \times \mathbb P^1$  
2) An open subset $U\subset X$ of an affine variety $X$ of positive dimension is never projective because the regular functions on $U$ obtained by restriction from  the regular functions on  $X$ separate the points of $U$ while the regular functions on a projective variety are constant. 
3) As for the ring of regular functions on $U=X\setminus Y$ in the case $\operatorname {codim} _X(Y)\geq 2$, it is easy to determine under a mild hypothesis (cf. Matsumura's Commutative Algebra, theorem 38, page 124):     
If  $X$ is normal, then  the restriction map $\mathcal O(X)\to \mathcal O(U)$ is bijective 
This is the algebraic version of an extraordinary result discovered in 1906 by Hartogs in the analytic setting. 
A: Try thinking about a simple case. Let $V=\mathbb{C}^2$, an affine variety with co-ordinates $x,y$. Then $\mathbb{C}^2-\{0\}=V_{x,y}$ in your notation. Show that $\mathbb{C}[V_{x,y}]=\mathbb{C}[x,y]$, thus showing that $V_{x,y}$ is not affine and answering your 3 negatively. For 2, no quasi affine varieties are projective except in the trivial case of finitely many points. Similarly, 4 can occur too, in the sense the coordinate ring of suitable quasi affine open sets can be say, non-Noetherian.
