I think that it is highly unlikely that there exists a separification functor. What does exist is the following:
Theorem (Raynaud-Gruson): Let $S$ be a base scheme and work relative to $S$. Given a non-separated scheme $X$ of finite type, there is a blow-up (a proper birational morphism) $X'\to X$ such that $X'$ admits an étale morphism to a projective scheme $Z$ (in particular a separated scheme).
Note that there are non-separated schemes which do not even admit a quasi-finite morphism onto a separated scheme (e.g. take $\mathbf A^2$ with a double origin and blow-up one of the origins).
The Theorem is false as stated for non-locally separated algebraic spaces. There are 3 different solutions to this:
A) Take an alteration instead of a modification.
B) Replace étale with quasi-finite flat.
C) Allow $Z$ to be a proper stack with finite diagonal.
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.
Best Answer
If $f_\ast$ has a right adjoint, it must preserve colimits and hence be right-exact. Thus a necessary condition is that the higher derived functors vanish. In particular, when everything is affine and we have $X=\operatorname{Spec} B$ and $Y=\operatorname{Spec} A$, then I believe the adjoint exists and is given by $M \mapsto \operatorname{Hom}_A(B,M)$.