Is there a result in computability theory that does not relativize?
Higman's Embedding Theorem: The finitely generated computably presented groups are precisely the finitely generated subgroups of finitely presented groups. Furthermore, every computably presented group (even ones countably generated) is a subgroup of a finitely presented group.

Note that this statement could relativize to: "The $O$-computably presented groups (with some oracle $O$) are precisely the finitely generated subgroups of finitely presented groups," but it does not, as one can prove that for some uncomputable $O$ there are $O$-computably presented groups that are not computably presented. 

Indeed, I think any non-relativizing result of computability theory must have something of this flavor, as some part of the result or its proof must somehow "nail down" true computability from computability with an oracle $O$. In this case, it is the finiteness which nails down "actual computability." Note that, as Scott Aaronson asked for, this result is invariant to any of the usual models of computation (Turing machine, RAM, etc), but does not relativize (again, because all of the usual models of "actual" computation share some common "finiteness property").

On the other hand, one might argue that this "doesn't count" for this question, as it is more akin to a definition of computability using groups than it is a "result of computability theory." On the other other hand, it is a definition of computability that is robust to model yet that doesn't relativize. (In contrast to, say, Kleene's characterization of the computable functions which easily relativizes, by simply adding the characteristic function of your oracle to the generating set of functions. There seems to be no analogous operation for groups in the context of Higman Embedding.)