Fix a field $K$ with absolute Galois group $G$. By an isogeny theorem over $K$, I mean the statement that the map $\operatorname{Hom}(A,B)\otimes\mathbb{Z}_l \to \operatorname{Hom}_G(T_l A, T_l B)$ is an isomorphism, where $A,B$ are abelian varieties over $K$, and $T_l A$ is the Tate module of $A$. Such a statement was proved for finite fields by Tate, for global function fields by Zarhin and for number fields by Faltings.
I'm interested in the case where $K$ is a $p$-adic field. The statement is then generally false, but is sometimes true. It holds e.g. when $A,B$ are elliptic curves with bad reduction and $l = p$ (Serre) or when $A,B$ have the same (good) reduction and again $l=p$ (Serre-Tate). It certainly fails if $A,B$ have good reduction and $l \ne p$.
What I don't have is a clear picture of the various possibilities for the reductions and for which cases the statement holds or fails. So that's the question.
Best Answer
I believe for $l\ne p$ the theorem is 'always' false, in the sense that for every positive-dimensional abelian variety $A$ one can find many $B$s for which your map is not onto, independently of the reduction type:
Fix $A$ and take any non-constant family of abelian varieties in which $A$ is a fibre, e.g. some neighbourhood of $A$ in the moduli space (choosing a polarization or whatever). Then all nearby fibers $A'$ in the family have the same $l$-adic representation, $V_l A\cong V_l A'$. This is a special case of a very general theorem of Mark Kisin that $l$-adic representations are locally constant in families in the $p$-adic topology (Math. Z. 230, 569-593 (1999), http://www.springerlink.com/content/44x7kn31jga7rlf7/, esp. lines 4-5 from the bottom). So there are uncountably many abelian varieties with the same $l$-adic representation as $A$, but only countably many isogenous ones, contradicting surjectivity.
This is just an extension of what you mentioned in the question: in the case of good reduction, $V_l A$ is a trivial inertia module so it is determined as a Galois representation by the characteristic polynomial of Frobenius. So, e.g., in a family of elliptic curves if $E$ and $E'$ are close enough so that they have the same reduction mod $p$, then $V_l E\cong V_l E'$; but, of course, there are uncountably many such 'nearby' $j$-invariants in any non-isotrivial family, so for any $E$ you can find lots of $E'$s for which the theorem fails.
(I don't know what happens for $l=p$, so this is only a partial answer, but it won't fit in a comment box.)