Early in the history of recursion theory, the realization that all known proofs in the subject could be relativized in the manner you indicate led Hartley Rogers to make what is called the homogeneity conjecture.

Let $\mathcal{D}$ be the structure of the Turing degrees with the partial order of Turing reducibility $\leq_T$. Let $\mathcal{D}(\geq \mathbf{x})$ be the structure of the Turing degrees that are $\geq_T \mathbf{x}$ also with the partial order $\leq_T$. The homogeneity conjecture says that for any Turing degree $\mathbf{x}$, $\mathcal{D}$ is isomorphic to $\mathcal{D}(\geq \mathbf{x})$.

Richard Shore refuted the homogeneity conjecture in an elegant 1979 paper -- it's only one page long (though it relies on earlier work coding models of arithmetic in $\mathcal{D}$). A couple years later, Harrington and Shore showed that you can do even better. Not only are there Turing degrees $\mathbf{x}$ for which $\mathcal{D}$ and $\mathcal{D}(\geq \mathbf{x})$ aren't isomorphic as structures, there are Turing degrees $\mathbf{x}$ so that $\mathcal{D}$ and $\mathcal{D}(\geq \mathbf{x})$ aren't elementarily equivalent.

So this means that there is a first order sentence $\varphi$ in the language only containing $\leq_T$ which is true about the Turing degrees, but which is false if you relativize the sentence to the Turing degrees $\geq_T \mathbf{x}$ for some $\mathbf{x}$ (and of course, working in the Turing degrees $\geq_T \mathbf{x}$ is equivalent to giving all Turing machines access to $\mathbf{x}$ as an oracle). Hence, the proof that $\varphi$ is true can't be relativized to $\mathbf{x}$.

A somewhat misleading and oversimplified explanation of why this occurs is that you can code models of arithmetic into $\mathcal{D}$ (or $\mathcal{D}(\geq \mathbf{x})$) which can then interpret unrelativized concepts like being arithmetically definable or hyperarithmetically definable.

A particularly spectacular form of this phenomenon would occur if Slaman and Woodin's biinterpertability conjecture is true. The conjecture says that the following relation (on $\overrightarrow{\mathbf{p}}$ and $\mathbf{d}$) is definable in $\mathcal{D}$: "$\overrightarrow{\mathbf{p}}$ codes a standard model of first order arithmetic and a real $X$ such that $X$ is of degree $\mathbf{d}$".

All that being said, just about every proof in recursion theory that I know of relativizes. I'd be very interested in a proof which doesn't relativize and doesn't factor through the coding machinery I've mentioned above.

By the way (and somewhat ironically), the Baker-Gill-Solovay theorem that you mentioned does itself relativize. The relativized version says that for any oracle $X$, there are oracles $A$ and $B$ so that $X$ is poly-time reducible to both $A$ and $B$, and $P^A = NP^A$ and $P^B \neq NP^B$. (We've relativized to $X$ here. The unrelativized result simply says that there are oracles $A$ and $B$ so that $P^A = NP^A$ and $P^B \neq NP^B$). Of course, the real point is that a proof that $P \neq NP$ can't use a technique that relativizes.