The abelian category of quasicoherent sheaves on a schemes determine the scheme. This is an old result of Gabriel ("des categories abeliennes" 1962), proved in full generality by Rosenberg. This means that, $\operatorname{QCoh}(X)$ does not only tell you the open subschemes of $X$ but also gives you the structure sheaf! I've known this result for some time but I had never looked at it in detail until today. I'll sketch what I have just learned hoping not to make big mistakes...

An abelian subcategory $B$ of an abelian category $A$ is said to be a **thick subcategory** if it is full and for any exact sequence in $A$

$$0\to M'\to M \to M''\to 0,$$

$M$ belongs to $B$ if and only if $M'$ and $M''$ do.

If $B$ is a thick subcategory of $A$ there is a well defined localization $A/B$, which is again an abelian category. $A/B$ has the same objects as $A$ and a morphism $f:M\to N$ in $A/B$ is an isomorphism if and only if $\ker f$ and $\operatorname{coker} f$ belong to $B$.

Let $T\colon A\to A/B$ be the localization functor. Then $B$ is said to be a **localizing subcategory** if $B$ is thick and $T$ has a right adjoint. The condition of being localizing can be rephrased only in terms of $A$ and $B$. see Gabriel's thesis above (proposition 4 in chapter III).

Finally, if $M$ is an object of $A$, we denote by $\langle M\rangle$; the smallest localizing subcategory containing $M$.

Now let $X$ be a scheme, $j\colon U \to X$ an open embedding and $i\colon Y\to X$ its closed complement. Then there are a bunch of adjunctions between the categories of quasicoherent sheaves of $U,X,Y$: $i_*\colon \operatorname{QCoh}(Y)\to \operatorname{QCoh}(X)$ has a left adjoint $i^*\colon \operatorname{QCoh}(X)\to \operatorname{QCoh}(Y)$ and a right adjoint $i_!\colon \operatorname{QCoh}(X) \to \operatorname{QCoh}(Y)$. On the other hand, the functor $j^*\colon \operatorname{QCoh}(X)\to \operatorname{QCoh}(U)$ has a left adjoint $j_!\colon \operatorname{QCoh}(U)\to \operatorname{QCoh}(X)$ and a right adjoint $j_*\colon \operatorname{QCoh}(U)\to \operatorname{QCoh}(X)$. This is sometimes called a recollement.

Let's assume that $X$ is Noetherian and let $A = \operatorname{QCoh}(X)$. We have an exact sequence of abelian categories

$$0 \to \operatorname{QCoh}(Y) \to A \to \operatorname{QCoh}(U) \to 0$$

in the sense that the category $\operatorname{QCoh}(Y)$ happens to be a localizing subcategory of $A$ and its quotient is identified with $\operatorname{QCoh}(U)$. The first map in the exact sequence is $i_*$ and the second $j^*$. Moreover, I think that $\operatorname{QCoh}(Y)$ is the smallest localizing subcategory of $\operatorname{QCoh}(X)$ containing $i_*O_Y$. Gabriel proves that there are no more such localizing subcategories, that is closed subschemes of $X$ correspond exactly to localizing subcategories $\langle M\rangle$ generated by a single coherent sheaf (i.e. Noetherian object in $A$). Moreover, irreducible closed subsets (the points in the underlying topological space of $X$) are given by localizing subcategories $\langle I\rangle$ for $I$ an indecomposable injective. We have described the points of $X$ and its closed sets in terms of only the category $A$, so we can recover the underlying topological space of $X$ from $A$.

In particular, an open subscheme $U$ of $X$ gives a complementary closed subscheme $Y$, which is in correspondence with a localizing subcategory $\langle M\rangle$ and, moreover, $\operatorname{QCoh}(U) = A/\langle M\rangle$. So, responding to the queston above, for any $f\colon U\to X$, $U$ is an open subscheme if and only if the kernel of $f^*\colon \operatorname{QCoh}(X) \to \operatorname{QCoh}(U)$ is a localizing subcategory of the form $\langle M\rangle$ for a coherent sheaf $M$.

Regarding the structure sheaf $O_X$ there is an isomorphism between $O_X(U)$ and the ring of endomorphisms of the identity functor on $\operatorname{QCoh}(U)$ (which happens to be $A/\operatorname{QCoh}(Y)$), so the structure sheaf can be recovered only in terms of the category $A$.

Finally, just say that there are other results in the spirit of reconstructing a scheme from some category of sheaves on it. This is the starting point for using such categories of sheaves as a definition of noncommutative scheme. There is more information on this entry in nlab.